Dimension 22

Theorem 4.12.1 The s.i.m.f. subgroups G of Sp₂₂(Q) are

# G |G| |Z(G)| L_min r.i.m.f.

supergroups [2,1,11] i[C4]¹¹₁ 2³⁰·3⁴·5²·7·11 3 [1,1,44] B22

1 _i[C₄]₁⊗A₁₁ 2¹²·3⁵·5²·7·11 10 [3²,2,264] A²₁₁ [2,2,11] ^√₋₃[C6]¹¹₁ 2¹⁹·3¹5·5²·7·11 1 [3¹¹,2,66] A¹¹₂

2 ^√₋₃[C₆]₁⊗A₁₁ 2¹¹·3⁶·5²·7·11 9 [3¹¹,4,396] A₁₁⊗A₂ 3 ^√₋₃[±U₅(2)◦C₃]₁₁ 2¹¹·3⁶·5·11 4 [2²·3,4,49896] S 4 ^√₋₂₃[±L₂(23)]₁₁ 2⁴·3·11·23 3 [23¹¹,12,1012]

[23¹¹,12,506] A⁽⁶⁾₂₂ where S := [±P SU₆(2).S₃]₂₂ <GL₂₂(Q).

Proof: By explicit calculations, one verifies that the above table is correct and yields s.i.m.f. groups. Further the r.i.m.f. supergroups are easily constructed since all s.i.m.f.

groups are uniform. It remains to show the completeness of the classification.

The group ^√₋₃[C₆]¹¹₁ is s.i.m.f. by Lemma 2.1.21 and _i[C₄]¹¹₁ fixes only three lattices.

One checks that it is s.i.m.f.. So we may now suppose thatG is s.p.i.m.f..

It follows from Corollary 4.1.2 thatE(G) is not trivial. Thus by Table 2.5.1, E(G) is conjugate to Alt₁₂, M₁₁, U₅(2) or L₂(23). In the latter two cases, E(G) is irreducible and B^o(E(G)) is already s.i.m.f..

If E(G)'Alt₁₂ then G contains a normal subgroup B := B^o(E(G))'A₁₁' ±S₁₂. ClearlyB is not self-centralizing. ThusGcontains an irreducible subgroup_i[C₄]₁⊗A₁₁ or^√₋₃[C₆]₁⊗A₁₁. Both groups are s.i.m.f..

If E(G)'M₁₁ then Out(E(G)) is trivial. Thus G contains an irreducible subgroup conjugate to ^√₋₃[C₆]₁⊗M₁₁ or _i[C₄]₁⊗M₁₁. The first group fixes 18 lattices and is only contained in^√₋₃[C₆]₁⊗A₁₁ and^√₋₃[C₆]¹¹₁ . The second group fixes 15 lattices and is only contained in _i[C₄]₁⊗A₁₁ and _i[C₄]¹¹₁ .

Invariant Forms

For each conjugacy class of s.p.i.m.f. matrix groups in GL_2n(Q) where 1≤n≤11 we give a symmetric positive definite form F and a skewsymmetric form S such that

Aut(Z^1×2n,{F, S}) = {g ∈GL_2n(Z)|gF g^tr =F and gSg^tr =S}

is a representative of that class.

Further the pairs (F, S) satisfy the following properties:

(a) det(F) = min{det(L, F⁰)|(L, F⁰)∈ Z(G)× F_>0(G) is integral} where G:= Aut(Z^1×2n,{F, S}).

(b) S·F⁻¹ generates the commuting algebra of Aut(Z^1×2n,{F, S}) and its minimal polynomial µ(S·F⁻¹, X) is one of

• X² +d for some squarefree d∈N.

• µ(ζ_k−ζ_k⁻¹, X) for some evenk ∈Z^≥6.

• µ(ζ₂₆+ζ₂₆³ +ζ₂₆⁹ , X).

• µ(√

k·(ζ_{`</sub>−ζ<sub>`}⁻¹), X) where (k, `)∈ {(2,10), (3,10), (3,16)}.

• µ(i+√

−3 +√

−5, X).

By Algorithm 2.3.3, these pairs (F, S) give an easy way to recover the conjugacy class of any given s.p.i.m.f. subgroup of GL_2n(Q) for 1 ≤n ≤11.

In some cases, tensoring such a pair (F, S) with a gram matrix g of some root lattice yields another s.p.i.m.f. automorphism group H. If the pair (F ⊗g, S⊗g) satisfies property (a) from above, then H is omitted in the list below, since these forms can easily be reconstructed from the name of H.

For example, tensoring the forms of_i[C₄]₁ with any gram matrix ofA₂ yields a pair of forms that satisfies the above properties. Thus the s.p.i.m.f. matrix group _i[C₄]₁⊗A₂ is omitted in the list below.

Similarly, tensoring the forms of _i[C₄]₁ with any gram matrix of A₅ yields a pair of forms whose automorphism group is conjugate to _i[C₄]₁⊗A₅. But these forms do not satisfy the above properties. Thus the list below contains some other forms for

i[C₄]₁⊗A₅.

119

Thepairs(F,S)aregivenasasinglematrixM.TheuppertriangularpartofMistheuppertriangularpartofS.Theotherentries ofMarethecorrespondingentriesofF. Forexample,21−33 −121−3 0−121 00−12describesthepair 0 BB @

2−100 −12−10 0−12−1 00−12

1 CC A,

0 BB @

01−33 −101−3 3−101 −33−10

1 CC A

.

A.1 Dimension 2

1:i[C4]1 11 01

2:√ −3[C6]1 23 −12

A.2 Dimension 4

1:i[(D8⊗C4).S3]2 210−2 −1211 0−120 0−102

3:√ −2[GL2(3)]2 202−2 −1202 0−120 0−102

4:∞,2[SL2(3)]1◦C3 2−12−2 −1211 0−122 0−102

5:ζ10[C10]1 21−33 −121−3 0−121 00−12

A.3 Dimension 6

1:√ −3[±31+2 +:SL2(3)]3 23−2000 −121000 0−121−21 00−1230 000−120 00−1002

2:√ −7[±L2(7)]3 470−770 −1470−77 −2−1470−7 1−2−1470 11−2−147 −211−2−14

A.4 Dimension 8

1:i[(21+4 +⊗C4).S6]4 200001−20 −1200001−1 0−120−1002 00−12010−1 000−12−100 0000−1200 00000−120 0000−1002

3:√ −2[∞,2[21+4 −.Alt5]2:2]4 200−220−10 −1201000−2 0−1200010 00−120−101 000−12010 0000−12−20 00000−120 0000−1002

4:√ −2[F4:2]4 2000002−2 −120002−20 0−1202−200 0−102−2000 00002000 0000−1200 00000−120 00000−102

5:√ −3[Sp4(3)◦C3]4 23−200000 −12100000 0−12−10002 00−12−1000 000−12−12−1 0000−12−30 00000−120 0000−1002 7:∞,2[SL2(5):2]2◦C3 423−3−10−22 −24−10−1−13−1 1−1413620 10−14−1−121 −1−111433−2 −21−211420 01−22−1243 2−101−2014

9:√ −5[√ 5,∞[SL2(5)]12− C4]4 20003−2−1−3 −1200−113−1 0−1200−313 00−1211−2−1 000−12100 0000−1200 00000−120 0000−1002

10:√ −5[√ 5,∞[SL2(5)]12+ C4]4 4500000−5 −24005500 1−14000−50 10−140550 −1−1114000 −21−211400 01−22−1240 2−101−2014

11:√ −5[C20:C4]4 2000−4111 −120011−41 0−1201111 00−121−411 00002000 0000−1200 00000−120 000000−12 12:√ −6[√ 2,∞[˜S4]12− C3]4 23−10−1000 −1201−22−12 0−1211−23−3 00−121−202 000−121−11 0000−12−2−1 00000−12−2 0000−1002

13:√ −6[√ 2,∞[˜S4]12+ C3]4 2000022−2 −1200−200−2 0−120−2004 0−10222−40 00002000 0000−1200 00000−120 00000−102

14:√ −6[D16

2 √ −3[C6]1]4 20000330 −1200−300−3 0020−3003 00−1203−30 00002000 0000−1200 00000020 000000−12

15:√ −7[2.Alt7]4 2−10−20004 −1230−2000 0−121020−2 00−123−400 000−1230−1 0000−1212 00000−120 0000−1002 16:√ −15[√ 5,∞[SL2(5)]12− C3]4 210020−60 −12−540−240 0−12−30000 00−12300−6 000−121−25 0000−1230 00000−120 0000−1002

17:√ −15[√ 5,∞[SL2(5)]12+ C3]4 40−55510100 −24−50−5−5−55 1−14−5−5000 10−145505 −1−11145−50 −21−211400 01−22−124−5 2−101−2014

18:√ −15[C30:C4]4 40006−966 −2400−96−96 0−2406−96−9 00−2466−96 −21004000 1−210−2400 01−210−240 001−200−24

19:√ 5,∞[SL2(5)]1◦C5 2−2100000 −12−210000 0−12−10000 00−120−1−12 000−122−1−2 0000−1221 00000−12−1 0000−1002

21:ζ16−ζ−1 16[QD32]2 10010100 01001010 00100101 00010010 00001001 00000100 00000010 00000001

A.5 Dimension 10

1:i[C4]1⊗A5 2000001−11−1 1200002−110 −10200−100−11 −10120000−10 10−1−12−10100 00000200−1−1 00000020−1−1 00000−1−1211 11−1−1011−13−2 11−1−10−10113

2:√ −3[±S4(3)◦C3]5 4000−30−3−300 −24003003−30 −224030030−3 −2224030300 −1−11040000−3 20−2−1−24−3003 −12020140−30 −1−11−12−2−240−3 −21200−2−1240 −2210−1−10124

3:√ −11[L2(11)]5 6000−11−110000 26−11110−1111000 −2−36001100011 −21−26000110−11 −3−22260011110 −31102611000 23−220−1611110 2−22−1−1−21600 2−2001−2−126−11 0211202016

A.6 Dimension 12

2:∞,3[±U3(3)]3◦C4 4−1−2−200000001 040001−100−1−10 104−110000−10−1 −221400000−1−2−1 01−2−14−100−2210 −21−21240000−11 0−20−20−141−122−1 1−22−1−2−224101−1 −200202−2−14−2−21 1000−2−201−2410 0212−1−1−200240 −20−20220−12−2−14

4:i[√ −3[±31+2 +:SL2(3)]3

2(2) @×⊃i[C4]1]6 4011−21−1−12100 241022101110 −11402240−1010 −10041−1010101 000−14−1011010 −10211421−211−1 −1−1002240−1010 100111041−1−2−1 2110−10−1140−1−1 −1−10101010400 0−1−121112124−1 000−10−10−11214

5:i[L2(7)2(2) ⊗i[C4]1]6 4100−1−2101−2−1−1 −141001101020 −214−104−110001 0−21401001001 120−240220011 0−10−104−1−100−21 1−1120−1421−101 00100104−1−20−1 −110−1001−140−10 2000201004−10 −1022−10201141 10−1−111112214

6:i[C4]1⊗A(2) 6 4000004−1−211−2 −140000−14−1−211 −2−14000−2−14−1−21 1−2−14001−2−14−1−2 11−2−14011−2−14−1 −211−2−14−211−2−14 000000400000 000000−140000 000000−2−14000 0000001−2−1400 00000011−2−140 000000−211−2−14

7:i[√ −7[±L2(7)]3

2(2) @×⊃i[C4]1]6 800−421−134−4−40 −2802341−4−121−3 −4481−111−3014−4 4−2−180−2−124−801 −2−31−28−302−220−1 −3430180−4−301−2 1333−428−13−31−2 3−4−300−4−3800−21 4−1000−1148000 −4210021−2−4843 4−1−44−2−1−100084 4−1−43−1−2−230−148

8:√ −2[±L2(7)·2]6 20000002−101−1 −120000−101−1−10 0−120002−10−111 00−1200−20101−1 000−12010−210−1 0000−121−120−10 000000200000 000000−120000 0000000−12000 00000000−1200 000000000−120 0000000000−12 9:√ −2[∞,2[SL2(5)]3:2]6 3120−2−1112−100 −13−100−30112−11 1131−2−1−22002−1 0−10322110−121 1−1−1−130001−1−11 1−1−1114301−1−11 111−11−14112−11 01100−1031111 1−1000−110300−1 −10−1−11−11−1030−1 0−1−1102−200−13−1 101−11−12110−13

10:√ −3[6.U4(3).2]6 403000030−300 040033000000 104300000030 −2214330−30000 01−2−1400000−30 −21−21243000−30 0−20−20−1400000 1−22−1−2−2243−303 −200202−2−14000 1000−2−201−2400 0212−1−1−200243 −20−20220−12−2−14

11:i[√ −3[±31+2 +:SL2(3)]3⊗∞,2[SL2(3)]1]6 4031−21−116122 243222123−11−2 −11422242−321−2 −1004112−3−2−30−3 000−14−30−11214 −1021146−1−2−111 −1−100224−2−1−2−12 100111041363 2110−10−114211 −1−10101010400 0−1−1211121243 000−10−10−11214 12:√ −3[±3.M10]6 83000−30−3−60−30 1803−6−333−6−30−12 208−3−603−6−3000 4−1−18003−300−30 022−280−360000 3−140480−60033 −4−3−3−3−3−4833633 −332−12−2−18−3−3−3−6 −2−23−4421180−36 −21−4−2−2−421−28−60 −3−4−410−13−11286 240−241−3022−28

13:√ −3[C6]1⊗M6,2 10000001566666 41000006156−6−66 441000066156−6−6 4−4410006−66156−6 4−4−441006−6−66156 44−4−441066−6−6615 −5−2−2−2−2−21000000 −2−5−222−24100000 −2−2−5−2224410000 −22−2−5−224−441000 −222−2−5−24−4−44100 −2−222−2−544−4−4410

14:∞,2[SL2(5)]3◦C3 41010−3−3−2−21−31 −14−1−2−232−1−2−132 2−14−11−11−1−31−4−1 −12−14−14310233 −22−11432−12123 −1110143−1−3−3−11 101−10141−1−111 0−1−11−1−1141−110 001201−1140−31 −111211112410 −1−10−10111114−1 1011−11101214

15:√ −5[i[C4]1

2− Alt5]6 1010−1010−100−1 010010−10−10−1−1 0011010−10100 0001−10−10−1010 0000100−10−100 00000110−10−11 000000100−100 00000001−1011 000000001000 00000000011−1 000000000010 000000000001

16:√ −5[i[C4]1

2+ Alt5]6 50000000−5000 250000000500 2250000000−50 2−22500500000 2−2−225000000−5 22−2−225050000 000000500000 000000250000 000000225000 0000002−22500 0000002−2−2250 00000022−2−225

17:√ −7[±L2(7)]3⊗ √ −7∞,3[˜S3]1 42−30002−1−41−40 0402−5−1202000 104300−2020−1−2 −2214130−14−202 01−2−14−2−2−20032 −21−2124300012 0−20−20−1400−200 1−22−1−2−224−310−5 −200202−2−14420 1000−2−201−240−2 0212−1−1−20024−5 −20−20220−12−2−14 19:√ −11[SL2(11)]6 111111111111 0111−1−111−11−1−1 001−111−11−11−11 0001111−11−1−1−1 00001−1111−1−11 0000011−1−111−1 0000001−11111 00000001−1−111 000000001−11−1 00000000011−1 000000000011 000000000001

20:√ −15[±3.Alt6·21]6 409600030−30−6 0400−33−600000 104300000030 −2214−33−6−36006 01−2−146000030 −21−21243000−30 0−20−20−1400060 1−22−1−2−2243−30−3 −200202−2−14000 1000−2−201−24−60 0212−1−1−20024−3 −20−20220−12−2−14

21:√ −15[±3.Alt6:21]6 815000−1501500−150 180−150−15151501500 208−150015015000 4−1−1800−151500−150 022−2801500000 3−140480000−1515 −4−3−3−3−3−48−15−15015−15 −332−12−2−181515150 −2−23−4421180150 −21−4−2−2−421−2800 −3−4−410−13−11280 240−241−3022−28 22:ζ10[C10]1⊗√ 5Alt5 4030000011−1−3 −2412−3000−1233 0−14−3131−2−1−4−2−1 01−241−11−11421 −20−104−1−120012 000−2−14000−110 0−2101−240−1−2−4−2 201−1−1004−3−10−1 −110020014124 1−1200−1122400 −10111−2201240 000010122224

23:ζ26+ζ

3 26+ζ

9 26

[C26:C3]3 21−13−32−10−13−32 −121−23−301000−1 0−122−23−1−23−310 00−122−203−3100 000−1221−3100−1 0000−120100−13 00000−12−11−13−3 000000−12−22−33 0000000−12−22−3 00000000−12−22 000000000−12−2 0000000000−12

A.7 Dimension 14

1:i[C4]1⊗E7 20−111001110010 02000000100−100 01200000011−21−1 01120000100−100 010120001−10000 01110200011−100 01111120001−110 010110021−1−10−11 −110111113−100−11 11011111130001 10000000−112000 −100000000−1−1200 11011111121−131 −100000000−1−11−12

2:i[U3(3)◦C4]7 3000−1−10−110−1000 −130−100−100−10001 1130−1−10−11−10111 1013001−31−1−100−1 010−130−11−10−1−101 010−113−31−1−100−10 −10−10003000−1−10−1 0−10000−130001−10 0100001−13−1−1001 −1000−101103000−1 0−1−100−10101300−1 1−1010−1001−11310 −110−11010111030 −10000−101000113

A.8 Dimension 16

2:i[(21+6 +⊗C4).Sp6(2)]8 400101021−1001000 24−10−10110−111100−1 2140010200−1−11000 1224−102200−100−1−2−1 21214200−11110000 1212241101011−10−2 −2−100−2−14000−200000 −2−100−2−124000000−1−1 −1−200−1−22240−100−1−21 −1100−1100−240−1112−1 01−11120−2−1042−1−10−1 0−1−10110−20−1240−101 −11−10−2−10201−1−24000 200−121−2−2−1−111−2400 200000−2−100000240 212120−2−1−1−1−1−10004

3:i[√ −3[Sp4(3)◦C3]4

2(2) @×⊃i[C4]1]8 60110−11−102013111 06−1301−1−1202−13111 −13600−11−220201−202 1−1−261301111−3−2−1−2−1 00−2360−31−1−1−10011−1 1−3−332602−20−2−1−11−1−2 −1330106−100001−1−21 3−1−2−1−12−16−3−1−32210−1 −2−2−21−120160−1−2−30−3−3 2−2−23320−10610−10−11 −2−2−21−120−1316−1−30−30 −130−10−30−2−2−2−161111 33120−110−11−116−1−12 31−21−11−13000136−11 3102−1−1−20−11−113360 −31211−21−31−1010−306

5:i[C4]1⊗M8,3 40000000422−2−1221 24000000240−1−212−1 20400000204−2−1212 −2−1−240000−2−1−242−1−20 −1−2−124000−1−2−1240−11 212−10400212−10400 221−2−1040221−2−1041 1−12010141−1201014 0000000040000000 0000000024000000 0000000020400000 00000000−2−1−240000 00000000−1−2−124000 00000000212−10400 00000000221−2−1040 000000001−1201014

6:∞,3[SL2(7)]4◦C4 6102100−12−1−110202 16101−10−10−200020−2 −3−16−20−1−21−100−20−2−1−1 231600−2−21−2−1−101−10 022361−2−11−2−1−1−22−10 −212206−2−112−1020−1−1 31−20−10602−3−101200 10−2−1−213620−11000−1 11−21−221061−1310−10 22−30−2−200063−12000 13012−12−1006−12120 1−2−2−2−2−3−1−202−26−1−103 33031021112−260−2−1 −2100−13113−1−1−1−16−1−1 3−1−10−203020111−160 121301−1−101101−116 7:√ −2[∞,2[21+6 −.O− 6(2)]4:2]8 4000000000000220 2400000000−220000 2140000000000200 1224000000022000 2121400000000002 1212240000020000 −2−100−2−1400200000−2 −2−100−2−124−200000−20 −1−200−1−222400−22000 −1100−1100−24−200000 01−11120−2−10402−200 0−1−10110−20−1240020 −11−10−2−10201−1−240−20 200−121−2−2−1−111−2400 200000−2−100000240 212120−2−1−1−1−1−10004

8:√ −2[21+6 +.(Alt8:2)]8 20000000−2100−11−11 −120000001−11−1101−2 0−120000002−100−101 00−1200001−21001−20 000−12000−11−1−11010 0000−12001−101−1−111 00000−120−11−1110−1−2 0000−10020011−2100 0000000020000000 00000000−12000000 000000000−1200000 0000000000−120000 00000000000−12000 000000000000−1200 0000000000000−120 000000000000−1002 10:√ −2[GL2(3)]2⊗ √ −2∞,5[SL2(5):2]2 40000000−131−1−4−111 −24000000−10−2−1001−1 0040000040−1−11−2−1−2 1−224000010−1−11−102 1−202400010−10−1003 −22−1−204000−1−13200−1 −2220−10402−1−4−311−30 −1−1−2−1−10−140−4232−1−21 0000000040000000 00000000−24000000 0000000000400000 000000001−2240000 000000001−2024000 00000000−22−1−20400 00000000−2220−1040 00000000−1−1−2−1−10−14

12:∞,2[21+6 −.O− 6(2)]4◦C3 402−1210011001000 243010−1101−3−31001 214−2210000−13−10−20 1224100000−300−101 212142001111000−2 121224−1101−2−13−300 −2−100−2−142−20−2000−20 −2−100−2−124−20−200013 −1−200−1−22242−100−10−1 −1100−1100−240−11−1−21 01−11120−2−104−21−3−2−1 0−1−10110−20−1242−10−3 −11−10−2−10201−1−24002 200−121−2−2−1−111−242−2 200000−2−100000240 212120−2−1−1−1−1−10004

13:√ −3[Sp4(3)◦C3]4⊗ √ −3∞,2[SL2(3)]1 6−6−3−303−33000−3−9−3−3−3 06−3303−33000−3333−3 −13600−33000003000 1−1−263303−33−3−3030−3 00−2360−33−3−3−3003−3−3 1−3−332600000−3−3−3−30 −133010630000330−3 3−1−2−1−12−163−330−6−303 −2−2−21−120166303033 2−2−23320−106−30−30−3−3 −2−2−21−120−131633030 −130−10−30−2−2−2−16333−3 33120−110−11−11633−6 31−21−11−130001363−3 3102−1−1−20−11−113360 −31211−21−31−1010−306

14:√ −3[C6]1⊗[(SL2(5)2 SL2(5)):2]8 8000000012−633−3−606 −48000000−612−30−333−3 2−28000003−312−33−6−60 20−28000030−3123363 −2−2228000−3−333123−3−6 −42−422800−63−6331260 02−44−248003−66−36123 4−202−40286−303−60312 −42−1−1120−280000000 2−4101−1−11−48000000 −11−41−12202−2800000 −101−4−1−1−2−120−280000 11−1−1−4−112−2−2228000 2−12−1−1−4−20−42−422800 0−12−21−2−4−102−44−2480 −210−120−1−44−202−4028 16:(∞,2[SL2(3)]1◦C3)⊗√ −3∞,5[SL2(5):2]2 12−6−6183−30−600−66030−3 612−1860−33−3−660060−6−6 −6−612−12−3633−666−66−963 −6−601230−336−6−66−63−6−3 −3−213126−6180−63606−6−6 130−2−61218−6−69−3−6−3−933 23−3−1−6612−1269003−9−3−3 011−3−66012−63−6−12−3−333 −4−6624−6−6−212−12603303 46−6−20−33−30126336−12−6 64−2−63−1−202212333−6−3 −6−4266−2−402−3−312−60−30 46−6−22−3−3−1−3332120−3−6 −3−63303−333−6−10−612612 422−2033−3−402−3−1−212−6 303−3033−3−1−21000612

17:∞,5[SL2(5)2 √ 5D10]4◦C3 4−2−22−222−20−1113011 −242200001000−2−20−1 −2242−2−2−2210200−132 2−2−240022−1−1002001 20−20422−211120−1−1−2 20−2024−2−22−10200−10 20−2222422011−11−2−1 −202−2−2−2−241−110−110−1 0−11−1−10−214−1023131 −122−1−1−101−1420−2011 1−200−10−1120400222 1−20000−102−2242130 1−2−22001−1−100043−20 −221−2−10−11−120−1−141−1 1−2−12−1−100−1−1012−141 −1−101−2−2−111−1000−114 18:∞,2[SL2(5)2(2) @×⊃D8]4◦C3 81100−44−10−44−1444−4 180−4−102−5−4−230−1−3−2−3 3480−5−3−2−4−4−33−5−2−22−3 4428403−5−4−330303−3 43148−3404−30−40300 4212383434404444 244143801−10−4−3−100 31412428412−60110 42444134803−5333−3 42331413284−14−44−4 43132422148−44−4−4−8 321244421348411−2 4123243214428−440 412214333443484−4 444324431443448−4 4131242434020448

19:√ −3[C60.(C4×C2)]8 800066−6−30−3−33603−3 08006063033−3−603−3 228030306−600003−6 2248300060660−3−30 −2−2−1−38000−3−30−30−63−3 −20−4−42830−126−6−60−306 −22120−1833330−6−63−6 −3−1−4−22218−330000−36 4022−1−4−1−18−633303−3 1−120−1−2−1−1−28−3−3330−3 1−100−20103−1800030 130010−203−1080000 −220−242−20−1−3−448000 4443−2−12−42−100083−6 1−3−3−312−1−1−1−21−4−2−180 −1322−300−2−330002−48

20:√ −5[((SL2(5)◦SL2(5)):2)

2− i[C4]1]8 200000003−10−201−12 −120000000111−101−2 0−1200000−200−112−30 00−12000010−12−1−100 000−120001−3200−111 0000−120001−20021−2 00000−120−212−1−2003 0000−1002−13−1−23−20−1 0000000020000000 00000000−12000000 000000000−1200000 0000000000−120000 00000000000−12000 000000000000−1200 0000000000000−120 000000000000−1002

21:√ −5[((SL2(5)◦SL2(5)):2)

2+ i[C4]1]8 4000000000000505 −240000005000−5−500 1−140000005000000 10−14000000000550 −1−111400000000050 −21−21140000050050 01−22−124000500000 2−101−201400055500 0000000040000000 00000000−24000000 000000001−1400000 0000000010−140000 00000000−1−1114000 00000000−21−211400 0000000001−22−1240 000000002−101−2014 22:√ −5[i[(D8⊗C4).S3]22+ i√ 5,∞[SL2(5)]1]8 4221−2122−310−43−2−22 241−2−3−213−23−3−5300−1 2140−6−14404−3−33−40−2 1224−5−22204−3−44−32−1 21214222−13−1−50−202 1212241−1−252−31−12−2 −2−100−2−14220−222−20−4 −2−100−2−1240204003−3 −1−200−1−22240−122−121 −1100−1100−2401−132−3 01−11120−2−104−21−3−21 0−1−10110−20−124−2103 −11−10−2−10201−1−24200 200−121−2−2−1−111−2402 200000−2−100000244 212120−2−1−1−1−1−10004

23:√ −5[i[(D8⊗C4).S3]2

2− i√ 5,∞[SL2(5)]1]8 8−5−50000−5000−50000 18005101050051055105 34805510005551010105 4428005−50−5−50−5055 431485000−5000500 421238−50−50000000 24414380−5−5005500 31412428050010550 4244413480−5−55555 4233141328050000 4313242214800000 32124442134805−50 4123243214428000 4122143334434800 4443244314434480 4131242434020448 24:√ −5[∞,5[SL2(5)2 √ 5D10]4:2]8 44004020−31−4−4−30−2−2 −2420−2−400−11−112002 −2240−2−402−12213033 2−2−240224000−2−20−1−1 20−204−4−400−1−3−1−20−11 20−20240−201−30−4−1−4−4 20−2222420−2−20−2−1−3−2 −202−2−2−2−2410211300 0−11−1−10−214210201−3 −122−1−1−101−142000−21 1−200−10−11204−1−300−2 1−20000−102−2240120 1−2−22001−1−1000410−4 −221−2−10−11−120−1−14−32 1−2−12−1−100−1−1012−14−1 −1−101−2−2−111−1000−114

25:√ −5[i[(D8⊗C4).S3]2

2 D10]8 4−204030−60−2040−204 −242−2−30−33202−2202−2 0−24003000−2000−200 0−2046−300−4200−4200 −21004−204030−60−204 1−211−242−2−30−33202−2 01−200−24003000−200 010−20−2046−300−4200 0000−21004−204030−6 00001−211−242−2−30−33 000001−200−2400300 0000010−20−2046−300 00000000−21004−204 000000001−211−242−2 0000000001−200−240 00000000010−20−204

26:√ −6[∞,2[21+4 −.Alt5]2

2(3) @×⊃A2]8 20000000−43−2010−10 −120000001−4301−10−1 0−120000021−43−1000 00−120000001−42011 000−12000−1−112−43−11 0000−120001001−412 00000−120100−113−4−2 0000−1002010−13−22−4 0000000020000000 00000000−12000000 000000000−1200000 0000000000−120000 00000000000−12000 000000000000−1200 0000000000000−120 000000000000−1002

27:√ −6[∞,2[21+4 −.Alt5]2

2(3) @×⊃ i∞,3[˜S3]1]8 4−22222020−2−22−22−20 24−2020−4−20−402−2400 2144−2202−22−2−22−2−20 1224−20−220−20−22200 21214−20420−4−40000 121224−200−220040−2 −2−100−2−140002−24−4−22 −2−100−2−124−440−24−4−20 −1−200−1−22240−4−44−404 −1100−1100−2444−222−4 01−11120−2−104−20422 0−1−10110−20−124−20−22 −11−10−2−10201−1−24002 200−121−2−2−1−111−24−2−2 200000−2−100000240 212120−2−1−1−1−1−10004 28:√ −6[∞,2[SL2(3)]1

2 i∞,3[SL2(9)]2]8 40033003300−33660 1403300−3000−33033 22433000030−30330 1104−3−300000−30−33−3 111140−3030000000 2121143303600030 12211140000−33033 11121114−300−6−3−33−3 1−11012014330−30−3−3 11120122140−6003−3 21210222124−6003−3 101102122224003−3 1102101111014000 2011220120001400 21112111101−11240 112022101−1111114

29:√ −6[F4

2 ∞,3[˜S3]1]8 4000000003−3030−30 040000003000000−6 −20400000−30000000 0−2−24000000000−360 000−24000300000−66 0000−2400000−3060−3 00000−240−3006−6000 000000−240−6006−300 20−10000040000000 020−1000004000000 −102−10000−20400000 0−1−12−10000−2−240000 000−12−100000−24000 0000−12−100000−2400 00000−12−100000−240 000000−12000000−24 30:√ −6[(F4⊗A2):2]8 40000000000000−60 −2400000000000006 −2140000000006000 1−2−24000000000−600 00−2140006000−6000 001−2−24000−6000600 00−21004000−600060 001−200−240006000−6 0000000040000000 00000000−24000000 00000000−21400000 000000001−2−240000 0000000000−214000 00000000001−2−2400 0000000000−210040 00000000001−200−24

32:√ −7[2.Alt7]4⊗√ −7∞,3[˜S3]1 1270007−7700−7−77000 512−7211407−700−7−70707 23121477007−140−1401407 −6−3−4120014−14014−7147000 −6−2161277−1400−777070 341−41120007−70−71470 1−3−641−4120−770014−147−14 336−4−4−40127−70−1407−714 −60−560−2−1−512777−7000 −6−6−402−11−5312147−7−140−14 156−31−106−3−412−1414070 1−3−4−21−26−2−15−2120−140−14 5−20−3−15−2−2−31−60120−70 −6−1−660−24−1620−2−61270 64−4−4−331−3−2−2121−112−7 −61−56622−662−22−46−112

33:√ −10[√ −2[GL2(3)]2

2+ √ −2√ 5,∞[SL2(5)]1]8 200000004−32−2−1−133 −12000000−35−3−13−2−10 0−12000000−25−2010−3 00−12000030−32−23−42 000−12000−1−1−1−13−23−2 0000−1200−312−1−33−13 00000−1203−1−113−21−4 0000−10020013−2−302 0000000020000000 00000000−12000000 000000000−1200000 0000000000−120000 00000000000−12000 000000000000−1200 0000000000000−120 000000000000−1002

34:√ −10[√ −2[GL2(3)]2

2− √ −2√ 5,∞[SL2(5)]1]8 4000000050−5100055 −24000000−5100−5−500−5 1−140000010−550−5−1005 10−140000−55050555 −1−11140000−51005−5−50 −21−211400−50000500 01−22−1240−55−55010100 2−101−201450−55−55510 0000000040000000 00000000−24000000 000000001−1400000 0000000010−140000 00000000−1−1114000 00000000−21−211400 0000000001−22−1240 000000002−101−2014 35:√ −10[√ −2[GL2(3)]2

2 D10]8 40040002000−80004 −24−4000−20008000−40 0−24−4020−20−808040−4 0−204−202080−80−4040 −2100400−40006000−8 1−211−244000−600080 01−200−244060−60−808 010−20−204−606080−80 0000−2100400−40002 00001−211−244000−20 000001−200−244020−2 0000010−20−204−2020 00000000−21004004 000000001−211−24−40 0000000001−200−24−4 00000000010−20−204

36:√ −15[((SL2(5)◦SL2(5)):2)

2− √ −3[C6]1]8 40000000−6−360−66−66 −24000000−36−30006−6 0−24000006−3−6300−66 00−2400000036−300−6 000−24000−600−36−963 0000−24006000−9636 00000−240−66−6063−6−6 0000−20046−66−636−6−6 −2100000040000000 1−2100000−24000000 01−2100000−2400000 001−2100000−240000 0001−2101000−24000 00001−2100000−2400 000001−2000000−240 0000100−20000−2004 37:√ −15[((SL2(5)◦SL2(5)):2)

2+ √ −3[C6]1]8 8000000000151515000 −4800000000−150−15151515 2−280000015−1501515000 20−280000150150−15−15015 −2−222800015−1515−150−15−150 −42−4228000150−15−15000 02−44−248001500−150015 4−202−402801501500150 −42−1−1120−280000000 2−4101−1−11−48000000 −11−41−12202−2800000 −101−4−1−1−2−120−280000 11−1−1−4−112−2−2228000 2−12−1−1−4−20−42−422800 0−12−21−2−4−102−44−2480 −210−120−1−44−202−4028

38:√ −15[(√ 5,∞[SL2(5)]1◦C3)2 √ 5D10]8 42−6−22262−63−1−53411 −2422404−4−1−2−420−4−4−3 −2246626−2−1−2−20−2−7−1−4 2−2−244−4−2−2−3−1004043 20−2046221−33243−10 20−20246−2−430−26234 20−222242−4−2−1−17147 −202−2−2−2−241−1−3−4−51−2−1 0−11−1−10−21454−41−11−7 −122−1−1−101−14−2−2−2−2−5−1 1−200−10−11204−2006−2 1−20000−102−2242−550 1−2−22001−1−10004304 −221−2−10−11−120−1−14−3−1 1−2−12−1−100−1−1012−143 −1−101−2−2−111−1000−114

39:ζ10[C10]1⊗√ 5((SL2(5)◦SL2(5)):2) 400000000221−13−2−2 −240000001−3000−301 −22400000−10−3−33−210 2−2−240000−2210040−2 20−2040001−213−31−10 20−2024003−132−30−20 20−222240−1021−220−2 −202−2−2−2−24−11−2−31−202 0−11−1−10−2142−1−12−212 −122−1−1−101−14−3−300−2−2 1−200−10−11204000−10 1−20000−102−2240110 1−2−22001−1−1000420−3 −221−2−10−11−120−1−14−21 1−2−12−1−100−1−1012−14−2 −1−101−2−2−111−1000−114

40:ζ10[C10]1⊗ √ 50∞,2[21+4 −.Alt5]2 410221−2−100000011 24−2010−3000−100010 214224−1−1−2222011−1 122402−11−2200010−2 212141−122000212−2 121224−2100−10211−2 −2−100−2−14−2−3231010−1 −2−100−2−124−12−10−101−1 −1−200−1−22242000−10−2 −1100−1100−240200−11 01−11120−2−104−2211−1 0−1−10110−20−1242100 −11−10−2−10201−1−24−202 200−121−2−2−1−111−242−1 200000−2−100000240 212120−2−1−1−1−1−10004 42:ζ10[C10]1⊗ √ 50∞,3[SL2(9)]2 40−13−130222321110 14−1101−3−1−2−20−1−1002 224303−1111110211 11041−10−30−1−2−21−111 111142−1−21−31−10113 212114−10−1111−2−3−2−1 122111400−11−11123 111211140−1−1−11021 1−1101201410−1−2−10−2 1112012214−1−30130 21210222124−1000−1 1011021222241−21−1 1102101111014021 20112201200014−11 21112111101−11241 112022101−1111114

44:ζ10[C10]1⊗ √ 50∞,3[SL2(3)2 C3]2 60−4101−130001060−6 060−10−33−300030−2−20 −1362−200−1−1−1002020 1−1−2630−20−20−33−20−3−1 00−236011−20−30−31−1−2 1−3−3326−30−1−1−33133−1 −1330106−20000−1−10−1 3−1−2−1−12−162−2113300 −2−2−21−120164003−203 2−2−23320−106−20−320−3 −2−2−21−120−1316−10−301 −130−10−30−2−2−2−16−2−3−30 33120−110−11−1160−3−3 31−21−11−13000136−30 3102−1−1−20−11−113360 −31211−21−31−1010−306 45:ζ10[C60.(C2×C2)]4 80330−404800500−7−5 08530000004300−33 228010−4400000−80−4 224860−4500−333−5−3−1 −2−2−1−3830−30−23−40050 −20−4−42841003−3−33−13 −22120−18−3−10−3−10013 −3−1−4−22218−45000−404 4022−1−4−1−180000−3−40 1−120−1−2−1−1−28−5550−2−3 1−100−20103−18−4−8000 130010−203−1084000 −220−242−20−1−3−448050 4443−2−12−42−10008−5−2 1−3−3−312−1−1−1−21−4−2−180 −1322−300−2−330002−48

46:ζ16−ζ−1 16[(D8⊗QD32).S3]4 20000000−1101−11−20 −12000000010−12−21−1 0−12000001−2100000 00−12000011−1−10100 000−12000−2−11010−1−1 0000−120010−11−1011 00000−120−111−1−11−11 0000−100211−11−1−111 0000000020000000 00000000−12000000 000000000−1200000 0000000000−120000 00000000000−12000 000000000000−1200 0000000000000−120 000000000000−1002

48:√ 2·(ζ10−ζ−1 10)[√ 2,∞[˜S4]1

2− √ 50ζ10[C10]1]4 200000002−2−13−10−10 −12000000−22−2−121−1−2 0−12000000−211−1−232 00−1200002−11−1−13−1−1 000−12000001−23−300 0000−1200−2011−12−1−2 00000−12022−32−1−120 0000−1002−12−21−21−13 0000000020000000 00000000−12000000 000000000−1200000 0000000000−120000 00000000000−12000 000000000000−1200 0000000000000−120 000000000000−1002

49:√ 2·(ζ10−ζ−1 10)[√ 2,∞[˜S4]1

2+ √ 50ζ10[C10]1]4 4020−202400−4−42−4−22 2420200200−2−20−2−42 214−2−4−64220−2−42−4−20 12240−20−2−24−2−2−2−2−22 21214−4224−4−4−20−204 1212240000−2−2−2−2−44 −2−100−2−14−2−2244000−4 −2−100−2−1240242000−4 −1−200−1−2224222004−4 −1100−1100−242−2−24−20 01−11120−2−1042−4204 0−1−10110−20−1240044 −11−10−2−10201−1−2400−2 200−121−2−2−1−111−2404 200000−2−100000242 212120−2−1−1−1−1−10004 50:√ 2·(ζ10−ζ−1 10)[D16

2 ζ10[C10]1]4 20000000−211−21−202 −120000002−30112−2−1 0−1200000−22−210−21−1 00−120000110−2−21−12 000020001−321−2−101 0000−120010−201−120 00000−120012−210−1−2 000000−12−20−10−2210 0000000020000000 00000000−12000000 000000000−1200000 0000000000−120000 0000000000002000 000000000000−1200 0000000000000−120 00000000000000−12

51:√ 3·(ζ10−ζ−1 10)[C60.(C2×C2)]4,(1,1,1) 200000004−3−12−13−2−2 −12000000−120−12−540 0−1200000−103−2−14−4−1 00−12000010−1−22−102 000−12000−12−44−3101 0000−1200−1−22−2301−3 00000−12000010−2−22 0000−10022−130−1−12−1 0000000020000000 00000000−12000000 000000000−1200000 0000000000−120000 00000000000−12000 000000000000−1200 0000000000000−120 000000000000−1002 52:√ 3·(ζ10−ζ−1 10)[C60.(C2×C2)]4,(i,1,1) 41−2−221−3−10213311−1 2412262−12156−200−4 214−421−1100−33−130−2 1224023242−22−213−5 21214−2224−2042−2−1−3 1212241−16−316−413−3 −2−100−2−140−1−2−2−1−2211 −2−100−2−124−21−3−2−20−20 −1−200−1−22240−6−42−1−31 −1100−1100−2431−52−1−3 01−11120−2−10411−23−1 0−1−10110−20−1242−14−2 −11−10−2−10201−1−24−4−3−1 200−121−2−2−1−111−2402 200000−2−100000241 212120−2−1−1−1−1−10004

53:√ 3·(ζ10−ζ−1 10)[C60.(C2×C2)]4,(1,1,−1) 40000000234304−2−2 −240000001−2−402−214 −22400000−1−2022−231 2−2−240000544−3−10−60 20−2040002−3−24−3003 20−202400−2210−26−10 20−22224042−1−1−21−24 −202−2−2−2−24−3−3−210−16−1 0−11−1−10−214−164−150−4 −122−1−1−101−14−400−1−10 1−200−10−11204−206−1−5 1−20000−102−2242320 1−2−22001−1−10004−2−4−3 −221−2−10−11−120−1−1422 1−2−12−1−100−1−1012−14−2 −1−101−2−2−111−1000−114

54:√ 3·(ζ10−ζ−1 10)[C60.(C2×C2)]4,(1,−1,1) 40000000−63−306−30−3 −24000000303−300−63 0−2400000−33−66−6333 00−2400000−36−63−330 000−2400060−630000 0000−2400−303−306−3−3 00000−2400−6330−300 0000−2004−33300−300 −2100000040000000 1−2100000−24000000 01−2100000−2400000 001−2100000−240000 0001−2101000−24000 00001−2100000−2400 000001−2000000−240 0000100−20000−2004

55:√ 3·(ζ10−ζ−1 10)[C60.(C2×C2)]4,(i,1,−1) 400042−400000−2−120 −24002−42−20000−12−11 0−240−42−4400002−12−2 0−2040−244000001−2−2 −21004000−2−1200000 1−211−2400−12−110000 01−200−2402−12−20000 010−20−20401−2−20000 0000−2100400042−40 00001−211−24002−42−2 000001−200−240−42−44 0000010−20−2040−244 00000000−21004000 000000001−211−2400 0000000001−200−240 00000000010−20−204 56:√ 3·(ζ10−ζ−1 10)[C60.(C2×C2)]4,(i,−1,1) 6000000−306603003 0600000−360003003 −13636330000−33003 1−1−26−3−6−3033000033 00−2360−3−300090330 1−3−33260−306600−330 −133010603−3033−306 3−1−2−1−12−16−33000−300 −2−2−21−120160−3−6−9−9−30 2−2−23320−10636−3−3−36 −2−2−21−120−13160−3−6−63 −130−10−30−2−2−2−163630 33120−110−11−116006 31−21−11−13000136−33 3102−1−1−20−11−113363 −31211−21−31−1010−306

57:√ 3·(ζ10−ζ−1 10)[C60.(C2×C2)]4,(1,−1,−1) 40000000060−30000 −24000000−60−300000 0−240000003060000 00−24000030−600000 −2100400000000−603 1−210−240000006030 01−210−24000000−30−6 001−200−240000−3060 0000000040000000 00000000−24000000 000000000−2400000 0000000000−240000 00000000−21004000 000000001−210−2400 0000000001−210−240 00000000001−200−24 58:√ 3·(ζ10−ζ−1 10)[C60.(C2×C2)]4,(i,−1,−1) 80−9−300030−3−3−6006−6 08−9−901803−183−3−60066 2280−60−3000−6660−30 2248−1500−3003−3−3609 −2−2−1−38−903−33−9−300−63 −20−4−42833009−9−9−69−9 −22120−18−6−63−12−90063 −3−1−4−22218−360−6−12030 4022−1−4−1−180−3339−33 1−120−1−2−1−1−28−666−30−6 1−100−20103−18006−30 130010−203−10801200 −220−242−20−1−3−4480−30 4443−2−12−42−1000860 1−3−3−312−1−1−1−21−4−2−180 −1322−300−2−330002−48

59:√ 3·(ζ16−ζ−1 16)[θ16,∞[Q32]12 C3]4 200000004−1−1−12−11−1 −12000000−230−12−2−1−1 0−12000000−220−12−1−1 00−120000−11−13−2−102 000−120000−31−24−11−3 0000−120002−21−33−22 00000−1202−1010−340 0000−1002030−1−2102 0000000020000000 00000000−12000000 000000000−1200000 0000000000−120000 00000000000−12000 000000000000−1200 0000000000000−120 000000000000−1002

60:√ 3·(ζ16−ζ−1 16)[D32

2 √ −3[C6]1]4 200300000000003−3 −123−30000000000−30 0020000300000000 00−1200−3000000000 0000203030000000 0000−120−30−3000000 0000002000000000 000000−1200000000 0000000020003−300 00000000−12000300 000000000020−3030 0000000000−12030−3 0000000000002000 000000000000−1200 0000000000000020 00000000000000−12

61:Q(i,√ 3,√ 5)[C60.C2]2 4−1−13−5−113−5−3−21−2212 1402−6−1−12−2−11−2−254−1 −1040−63−20−3−35−1−245−3 10041−23−230−2020−40 −100−1420−6043−4433−4 11−1004−12−3−1−15−5−103 −1−10−10−141110−23−1−12 −1000001431−430−5−31 −10112−111411−3511−3 11120−1−1−1142−5262−4 0110−11−2−2−1043−5021 −10−1021011−1−141−1−10 000−20110−10−1−1433−1 0120−1−111100−1−1403 12−10−12−11−10011041 −2−1−1−20−121−1−2−101−1−14 62:ζ32−ζ−1 32[QD64]2 1100000000000001 0110000000000000 0011000000000000 0001100000000000 0000110000000000 0000011000000000 0000001100000000 0000000110000000 0000000011000000 0000000001100000 0000000000110000 0000000000011000 0000000000001100 0000000000000110 0000000000000011 0000000000000001

63:ζ34[C34]1 2001−21−110000001−1 −1200−1110−100000−11 0−1210−10−1100000−11 00−1210000000001−1 000−1200000000000 0000−12001−1−110000 00000−120−111−10−110 000000−12001−101−10 0000000−120−111−100 00000000−120−10101 000000000−121−100−1 0000000000−12−1100 00000000000−12−100 000000000000−120−1 0000000000000−121 00000000000000−12

A.9 Dimension 18

6: ^√₋₁₉[±L2(19)]9

4: _i[(D₈⊗C₄).S₃]₂⊗A₅

8: _i[C₄]₁⊗A⁽³⁾₁₀

13: ∞,2[±U₅(2)]₅◦C₃

17: ^√₋₃[^√₋₁₁[±L₂(11)]₅²⁽³⁾@×⊃^√₋₃[C₆]₁]₁₀

21: ^√₋₇[2.M₂₂: 2]₁₀

25: _ζ10[±5¹⁺²₊ .Sp₂(5)]₅

3: ^√₋₃[±U₅(2)◦C₃]₁₁

LetG < GL_m(Q) be a matrix group and let N and H be any groups.

r.i.m.f. rational irreducible maximal finite s.i.m.f. symplectic irreducible maximal finite

s.p.i.m.f. symplectic primitive irreducible maximal finite Π(k) set of all primes dividing the integer k

Π(K, k)˜ set of primes needed for the m-parameter argument cf. Definition 2.2.10 Fq finite field with q elements

Q_P1,...,Pr quaternion algebra with center Q ramified only at the places P₁, . . . , P_r Qα,P1,...,Pr quaternion algebra with centerQ(α) ramified only at the placesP1, . . . , Pr

ZK maximal order of an algebraic number field K Cl(R) ideal class group of a Dedekind ring R

ζn primitive n-th root of unity θ_n ζ_n+ζ_n⁻¹

I_m m×m identity matrix

F(G) set ofG-invariant forms cf. Definition 2.1.1 F_sym(G) subset of symmetric G-invariant forms in F(G) F_sym(G) subset of skewsymmetric G-invariant forms in F(G) F>0(G) subset of positive definite forms in Fsym(G)

G enveloping algebra of G cf. Definition 2.1.1

End(G) endomorphism ring or commuting algebra C_Q^m×m(G) of G cf. Defini-tion 2.1.1

Z(Λ) set of Λ-invariant lattices in Q^1×m for some Z-order Λ⊂ Q^m×m cf. Defi-nition 2.1.4

145

Z(G) set ofG-invariant lattices in Q^1×m cf. Definition 2.1.4

det(L, F) determinant of the lattice Lin the Euclidean space (L⊗R, F)

Aut_K(L,F) group ofK-linear automorphisms of the latticeLwrt. the forms inF cf.

Definition 2.1.4

B^o(G) generalized Bravais group of Gcf. Definition 2.1.22

±G the group hG,−I_mi

O_p(H) largest normal p-subgroup of H

F(H) Fitting subgroup ofH, i.e. the largest normal nilpotent subgroup of H E(H) layer ofH, i.e. the central product of all components of H

F^∗(H) generalized Fitting subgroup of H (central product ofF(H) and E(H)) A_n, F₄, E_k (automorphism groups of) root lattices

C_n cyclic group of order n

S_n, Alt_n symmetric and alternating groups onn letters D2n dihedral group of order 2n

QD₂ⁿ quasidihedral group of order 2ⁿ

Q₂ⁿ (generalized) quaternion group of order 2ⁿ 2¹⁺²ⁿ₊ central product of n copies of D₈

2¹⁺²ⁿ₋ central product of Q₈ and n−1 copies ofD₈

p¹⁺²ⁿ₊ extraspecialp-group of order p¹⁺²ⁿ and exponent p (podd prime) p¹⁺²ⁿ₋ extraspecialp-group of order p¹⁺²ⁿ and exponent p² (podd prime) G₁⊗G₂ tensor product of the two matrix groups G₁ and G₂. See Section 2.4 for

an explanation of the symbols G₁⊗ G₂, G₁⊗

Q

G₂ and G₁◦G₂

G₁^2(p)⊗G₂ extension of G₁⊗G₂ by C₂. See Section 2.4 for an explanation of the symbolsG₁^2(p)⊗G₂,G₁^2(p)⊗

Q

G₂,G₁

2(p)

G₂,G₁

2(p)

QG₂,G₁^2(p)@×⊃G₂,G₁^2(p)@×⊃

Q

G₂,G₁^2(p)◦G₂, G₁^2(p)G₂ and G₁^2(p)@⊃G₂

N:H semidirect product, i.e. a split extension ofN byH N·H nonsplit extension of N byH

N.H any extension of N by H HoS_k wreath product ofH and S_k

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