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G = He3⋊3Q16  order 432 = 24·33

2nd semidirect product of He3 and Q16 acting via Q16/C4=C22

non-abelian, supersoluble, monomial

Aliases: He3⋊3Q16, C32⋊Dic12, C12.12S32, (C3×C12).7D6, (C3×C6).4D12, C32⋊4C8.S3, (C2×He3).12D4, He3⋊3C8.1C2, He3⋊3Q8.2C2, He3⋊4Q8.1C2, C4.4(C32⋊D6), C32⋊4Q8.2S3, C2.7(He3⋊3D4), C32⋊2(C3⋊Q16), (C4×He3).7C22, C6.32(C3⋊D12), C3.3(C32⋊3Q16), (C3×C6).7(C3⋊D4), SmallGroup(432,86)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3 — C4×He3 — He3⋊3Q16
Chief series
C1 — C3 — C32 — He3 — C2×He3 — C4×He3 — He3⋊3Q8 — He3⋊3Q16
Lower central
He3 — C2×He3 — C4×He3 — He3⋊3Q16
Upper central
C1 — C2 — C4

Generators and relations for He3⋊3Q16
 G = < a,b,c,d,e | a3=b3=c3=d8=1, e2=d4, ab=ba, cac-1=ab-1, ad=da, eae-1=a-1, bc=cb, dbd-1=ebe-1=b-1, dcd-1=c-1, ce=ec, ede-1=d-1 >

Subgroups: 387 in 75 conjugacy classes, 21 normal (all characteristic)
C1, C2, C3, C3, C4, C4, C6, C6, C8, Q8, C32, C32, Dic3, C12, C12, Q16, C3×C6, C3×C6, C3⋊C8, C24, Dic6, C3×Q8, He3, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, Dic12, C3⋊Q16, C2×He3, C3×C3⋊C8, C32⋊4C8, C3×Dic6, C32⋊4Q8, C32⋊C12, He3⋊3C4, C4×He3, C32⋊2Q16, C32⋊3Q16, He3⋊3C8, He3⋊3Q8, He3⋊4Q8, He3⋊3Q16
Quotients: C1, C2, C22, S3, D4, D6, Q16, D12, C3⋊D4, S32, Dic12, C3⋊Q16, C3⋊D12, C32⋊D6, C32⋊3Q16, He3⋊3D4, He3⋊3Q16

Character table of He3⋊3Q16

 class 123A3B3C3D4A4B4C6A6B6C6D8A8B12A12B12C12D12E12F12G12H12I12J24A24B24C24D
 size 1126612236362661218184661212123636363618181818
ρ111111111111111111111111111111    trivial
ρ21111111-111111-1-1111111-111-1-1-1-1-1    linear of order 2
ρ311111111-11111-1-11111111-1-11-1-1-1-1    linear of order 2
ρ41111111-1-1111111111111-1-1-1-11111    linear of order 2
ρ5222-12-120-22-12-100222-1-1-101100000    orthogonal lifted from D6
ρ6222222-200222200-2-2-2-2-2-200000000    orthogonal lifted from D4
ρ72222-1-120022-1-1222-1-12-1-10000-1-1-1-1    orthogonal lifted from S3
ρ82222-1-120022-1-1-2-22-1-12-1-100001111    orthogonal lifted from D6
ρ9222-12-12022-12-100222-1-1-10-1-100000    orthogonal lifted from S3
ρ102222-1-1-20022-1-100-211-2110000√3-√3√3-√3    orthogonal lifted from D12
ρ112222-1-1-20022-1-100-211-2110000-√3√3-√3√3    orthogonal lifted from D12
ρ122-22222000-2-2-2-2√2-√20000000000√2√2-√2-√2    symplectic lifted from Q16, Schur index 2
ρ132-22222000-2-2-2-2-√2√20000000000-√2-√2√2√2    symplectic lifted from Q16, Schur index 2
ρ142-222-1-1000-2-211-√2√20-√3√30√3-√30000ζ83ζ3+ζ8ζ3+ζ8ζ87ζ3+ζ87+ζ85ζ3ζ83ζ32+ζ83+ζ8ζ32ζ87ζ32+ζ85ζ32+ζ85    symplectic lifted from Dic12, Schur index 2
ρ152-222-1-1000-2-211√2-√20-√3√30√3-√30000ζ83ζ32+ζ83+ζ8ζ32ζ87ζ32+ζ85ζ32+ζ85ζ83ζ3+ζ8ζ3+ζ8ζ87ζ3+ζ87+ζ85ζ3    symplectic lifted from Dic12, Schur index 2
ρ162-222-1-1000-2-211-√2√20√3-√30-√3√30000ζ87ζ3+ζ87+ζ85ζ3ζ83ζ3+ζ8ζ3+ζ8ζ87ζ32+ζ85ζ32+ζ85ζ83ζ32+ζ83+ζ8ζ32    symplectic lifted from Dic12, Schur index 2
ρ172-222-1-1000-2-211√2-√20√3-√30-√3√30000ζ87ζ32+ζ85ζ32+ζ85ζ83ζ32+ζ83+ζ8ζ32ζ87ζ3+ζ87+ζ85ζ3ζ83ζ3+ζ8ζ3+ζ8    symplectic lifted from Dic12, Schur index 2
ρ18222-12-1-2002-12-100-2-2-21110-√-3√-300000    complex lifted from C3⋊D4
ρ19222-12-1-2002-12-100-2-2-21110√-3-√-300000    complex lifted from C3⋊D4
ρ20444-2-21-4004-2-2100-4222-1-100000000    orthogonal lifted from C3⋊D12
ρ21444-2-214004-2-21004-2-2-21100000000    orthogonal lifted from S32
ρ224-44-24-2000-42-420000000000000000    symplectic lifted from C3⋊Q16, Schur index 2
ρ234-44-2-21000-422-10002√3-2√30√3-√300000000    symplectic lifted from C32⋊3Q16, Schur index 2
ρ244-44-2-21000-422-1000-2√32√30-√3√300000000    symplectic lifted from C32⋊3Q16, Schur index 2
ρ2566-3000620-300000-300000-100-10000    orthogonal lifted from C32⋊D6
ρ2666-30006-20-300000-30000010010000    orthogonal lifted from C32⋊D6
ρ2766-3000-600-300000300000-√-300√-30000    complex lifted from He3⋊3D4
ρ2866-3000-600-300000300000√-300-√-30000    complex lifted from He3⋊3D4
ρ2912-12-600000060000000000000000000    symplectic faithful, Schur index 2

Smallest permutation representation of He3⋊3Q16
►On 144 points
Generators in S144
(9 44 58)(10 45 59)(11 46 60)(12 47 61)(13 48 62)(14 41 63)(15 42 64)(16 43 57)(49 95 137)(50 96 138)(51 89 139)(52 90 140)(53 91 141)(54 92 142)(55 93 143)(56 94 144)(65 126 117)(66 127 118)(67 128 119)(68 121 120)(69 122 113)(70 123 114)(71 124 115)(72 125 116)(73 85 102)(74 86 103)(75 87 104)(76 88 97)(77 81 98)(78 82 99)(79 83 100)(80 84 101)

(1 132 107)(2 108 133)(3 134 109)(4 110 135)(5 136 111)(6 112 129)(7 130 105)(8 106 131)(9 44 58)(10 59 45)(11 46 60)(12 61 47)(13 48 62)(14 63 41)(15 42 64)(16 57 43)(17 30 37)(18 38 31)(19 32 39)(20 40 25)(21 26 33)(22 34 27)(23 28 35)(24 36 29)(49 95 137)(50 138 96)(51 89 139)(52 140 90)(53 91 141)(54 142 92)(55 93 143)(56 144 94)(65 126 117)(66 118 127)(67 128 119)(68 120 121)(69 122 113)(70 114 123)(71 124 115)(72 116 125)(73 85 102)(74 103 86)(75 87 104)(76 97 88)(77 81 98)(78 99 82)(79 83 100)(80 101 84)

(1 68 100)(2 101 69)(3 70 102)(4 103 71)(5 72 104)(6 97 65)(7 66 98)(8 99 67)(9 25 140)(10 141 26)(11 27 142)(12 143 28)(13 29 144)(14 137 30)(15 31 138)(16 139 32)(17 41 95)(18 96 42)(19 43 89)(20 90 44)(21 45 91)(22 92 46)(23 47 93)(24 94 48)(33 59 53)(34 54 60)(35 61 55)(36 56 62)(37 63 49)(38 50 64)(39 57 51)(40 52 58)(73 134 114)(74 115 135)(75 136 116)(76 117 129)(77 130 118)(78 119 131)(79 132 120)(80 113 133)(81 105 127)(82 128 106)(83 107 121)(84 122 108)(85 109 123)(86 124 110)(87 111 125)(88 126 112)

(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)(65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80)(81 82 83 84 85 86 87 88)(89 90 91 92 93 94 95 96)(97 98 99 100 101 102 103 104)(105 106 107 108 109 110 111 112)(113 114 115 116 117 118 119 120)(121 122 123 124 125 126 127 128)(129 130 131 132 133 134 135 136)(137 138 139 140 141 142 143 144)

(1 29 5 25)(2 28 6 32)(3 27 7 31)(4 26 8 30)(9 100 13 104)(10 99 14 103)(11 98 15 102)(12 97 16 101)(17 110 21 106)(18 109 22 105)(19 108 23 112)(20 107 24 111)(33 131 37 135)(34 130 38 134)(35 129 39 133)(36 136 40 132)(41 86 45 82)(42 85 46 81)(43 84 47 88)(44 83 48 87)(49 115 53 119)(50 114 54 118)(51 113 55 117)(52 120 56 116)(57 80 61 76)(58 79 62 75)(59 78 63 74)(60 77 64 73)(65 139 69 143)(66 138 70 142)(67 137 71 141)(68 144 72 140)(89 122 93 126)(90 121 94 125)(91 128 95 124)(92 127 96 123)
 
G:=sub<Sym(144)| (9,44,58)(10,45,59)(11,46,60)(12,47,61)(13,48,62)(14,41,63)(15,42,64)(16,43,57)(49,95,137)(50,96,138)(51,89,139)(52,90,140)(53,91,141)(54,92,142)(55,93,143)(56,94,144)(65,126,117)(66,127,118)(67,128,119)(68,121,120)(69,122,113)(70,123,114)(71,124,115)(72,125,116)(73,85,102)(74,86,103)(75,87,104)(76,88,97)(77,81,98)(78,82,99)(79,83,100)(80,84,101), (1,132,107)(2,108,133)(3,134,109)(4,110,135)(5,136,111)(6,112,129)(7,130,105)(8,106,131)(9,44,58)(10,59,45)(11,46,60)(12,61,47)(13,48,62)(14,63,41)(15,42,64)(16,57,43)(17,30,37)(18,38,31)(19,32,39)(20,40,25)(21,26,33)(22,34,27)(23,28,35)(24,36,29)(49,95,137)(50,138,96)(51,89,139)(52,140,90)(53,91,141)(54,142,92)(55,93,143)(56,144,94)(65,126,117)(66,118,127)(67,128,119)(68,120,121)(69,122,113)(70,114,123)(71,124,115)(72,116,125)(73,85,102)(74,103,86)(75,87,104)(76,97,88)(77,81,98)(78,99,82)(79,83,100)(80,101,84), (1,68,100)(2,101,69)(3,70,102)(4,103,71)(5,72,104)(6,97,65)(7,66,98)(8,99,67)(9,25,140)(10,141,26)(11,27,142)(12,143,28)(13,29,144)(14,137,30)(15,31,138)(16,139,32)(17,41,95)(18,96,42)(19,43,89)(20,90,44)(21,45,91)(22,92,46)(23,47,93)(24,94,48)(33,59,53)(34,54,60)(35,61,55)(36,56,62)(37,63,49)(38,50,64)(39,57,51)(40,52,58)(73,134,114)(74,115,135)(75,136,116)(76,117,129)(77,130,118)(78,119,131)(79,132,120)(80,113,133)(81,105,127)(82,128,106)(83,107,121)(84,122,108)(85,109,123)(86,124,110)(87,111,125)(88,126,112), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104)(105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128)(129,130,131,132,133,134,135,136)(137,138,139,140,141,142,143,144), (1,29,5,25)(2,28,6,32)(3,27,7,31)(4,26,8,30)(9,100,13,104)(10,99,14,103)(11,98,15,102)(12,97,16,101)(17,110,21,106)(18,109,22,105)(19,108,23,112)(20,107,24,111)(33,131,37,135)(34,130,38,134)(35,129,39,133)(36,136,40,132)(41,86,45,82)(42,85,46,81)(43,84,47,88)(44,83,48,87)(49,115,53,119)(50,114,54,118)(51,113,55,117)(52,120,56,116)(57,80,61,76)(58,79,62,75)(59,78,63,74)(60,77,64,73)(65,139,69,143)(66,138,70,142)(67,137,71,141)(68,144,72,140)(89,122,93,126)(90,121,94,125)(91,128,95,124)(92,127,96,123)>;
 
G:=Group( (9,44,58)(10,45,59)(11,46,60)(12,47,61)(13,48,62)(14,41,63)(15,42,64)(16,43,57)(49,95,137)(50,96,138)(51,89,139)(52,90,140)(53,91,141)(54,92,142)(55,93,143)(56,94,144)(65,126,117)(66,127,118)(67,128,119)(68,121,120)(69,122,113)(70,123,114)(71,124,115)(72,125,116)(73,85,102)(74,86,103)(75,87,104)(76,88,97)(77,81,98)(78,82,99)(79,83,100)(80,84,101), (1,132,107)(2,108,133)(3,134,109)(4,110,135)(5,136,111)(6,112,129)(7,130,105)(8,106,131)(9,44,58)(10,59,45)(11,46,60)(12,61,47)(13,48,62)(14,63,41)(15,42,64)(16,57,43)(17,30,37)(18,38,31)(19,32,39)(20,40,25)(21,26,33)(22,34,27)(23,28,35)(24,36,29)(49,95,137)(50,138,96)(51,89,139)(52,140,90)(53,91,141)(54,142,92)(55,93,143)(56,144,94)(65,126,117)(66,118,127)(67,128,119)(68,120,121)(69,122,113)(70,114,123)(71,124,115)(72,116,125)(73,85,102)(74,103,86)(75,87,104)(76,97,88)(77,81,98)(78,99,82)(79,83,100)(80,101,84), (1,68,100)(2,101,69)(3,70,102)(4,103,71)(5,72,104)(6,97,65)(7,66,98)(8,99,67)(9,25,140)(10,141,26)(11,27,142)(12,143,28)(13,29,144)(14,137,30)(15,31,138)(16,139,32)(17,41,95)(18,96,42)(19,43,89)(20,90,44)(21,45,91)(22,92,46)(23,47,93)(24,94,48)(33,59,53)(34,54,60)(35,61,55)(36,56,62)(37,63,49)(38,50,64)(39,57,51)(40,52,58)(73,134,114)(74,115,135)(75,136,116)(76,117,129)(77,130,118)(78,119,131)(79,132,120)(80,113,133)(81,105,127)(82,128,106)(83,107,121)(84,122,108)(85,109,123)(86,124,110)(87,111,125)(88,126,112), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104)(105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128)(129,130,131,132,133,134,135,136)(137,138,139,140,141,142,143,144), (1,29,5,25)(2,28,6,32)(3,27,7,31)(4,26,8,30)(9,100,13,104)(10,99,14,103)(11,98,15,102)(12,97,16,101)(17,110,21,106)(18,109,22,105)(19,108,23,112)(20,107,24,111)(33,131,37,135)(34,130,38,134)(35,129,39,133)(36,136,40,132)(41,86,45,82)(42,85,46,81)(43,84,47,88)(44,83,48,87)(49,115,53,119)(50,114,54,118)(51,113,55,117)(52,120,56,116)(57,80,61,76)(58,79,62,75)(59,78,63,74)(60,77,64,73)(65,139,69,143)(66,138,70,142)(67,137,71,141)(68,144,72,140)(89,122,93,126)(90,121,94,125)(91,128,95,124)(92,127,96,123) );
 
G=PermutationGroup([[(9,44,58),(10,45,59),(11,46,60),(12,47,61),(13,48,62),(14,41,63),(15,42,64),(16,43,57),(49,95,137),(50,96,138),(51,89,139),(52,90,140),(53,91,141),(54,92,142),(55,93,143),(56,94,144),(65,126,117),(66,127,118),(67,128,119),(68,121,120),(69,122,113),(70,123,114),(71,124,115),(72,125,116),(73,85,102),(74,86,103),(75,87,104),(76,88,97),(77,81,98),(78,82,99),(79,83,100),(80,84,101)], [(1,132,107),(2,108,133),(3,134,109),(4,110,135),(5,136,111),(6,112,129),(7,130,105),(8,106,131),(9,44,58),(10,59,45),(11,46,60),(12,61,47),(13,48,62),(14,63,41),(15,42,64),(16,57,43),(17,30,37),(18,38,31),(19,32,39),(20,40,25),(21,26,33),(22,34,27),(23,28,35),(24,36,29),(49,95,137),(50,138,96),(51,89,139),(52,140,90),(53,91,141),(54,142,92),(55,93,143),(56,144,94),(65,126,117),(66,118,127),(67,128,119),(68,120,121),(69,122,113),(70,114,123),(71,124,115),(72,116,125),(73,85,102),(74,103,86),(75,87,104),(76,97,88),(77,81,98),(78,99,82),(79,83,100),(80,101,84)], [(1,68,100),(2,101,69),(3,70,102),(4,103,71),(5,72,104),(6,97,65),(7,66,98),(8,99,67),(9,25,140),(10,141,26),(11,27,142),(12,143,28),(13,29,144),(14,137,30),(15,31,138),(16,139,32),(17,41,95),(18,96,42),(19,43,89),(20,90,44),(21,45,91),(22,92,46),(23,47,93),(24,94,48),(33,59,53),(34,54,60),(35,61,55),(36,56,62),(37,63,49),(38,50,64),(39,57,51),(40,52,58),(73,134,114),(74,115,135),(75,136,116),(76,117,129),(77,130,118),(78,119,131),(79,132,120),(80,113,133),(81,105,127),(82,128,106),(83,107,121),(84,122,108),(85,109,123),(86,124,110),(87,111,125),(88,126,112)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64),(65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88),(89,90,91,92,93,94,95,96),(97,98,99,100,101,102,103,104),(105,106,107,108,109,110,111,112),(113,114,115,116,117,118,119,120),(121,122,123,124,125,126,127,128),(129,130,131,132,133,134,135,136),(137,138,139,140,141,142,143,144)], [(1,29,5,25),(2,28,6,32),(3,27,7,31),(4,26,8,30),(9,100,13,104),(10,99,14,103),(11,98,15,102),(12,97,16,101),(17,110,21,106),(18,109,22,105),(19,108,23,112),(20,107,24,111),(33,131,37,135),(34,130,38,134),(35,129,39,133),(36,136,40,132),(41,86,45,82),(42,85,46,81),(43,84,47,88),(44,83,48,87),(49,115,53,119),(50,114,54,118),(51,113,55,117),(52,120,56,116),(57,80,61,76),(58,79,62,75),(59,78,63,74),(60,77,64,73),(65,139,69,143),(66,138,70,142),(67,137,71,141),(68,144,72,140),(89,122,93,126),(90,121,94,125),(91,128,95,124),(92,127,96,123)]])
 

Matrix representation of He3⋊3Q16 ►in GL10(𝔽73)

07200000000
17200000000
00072000000
00172000000
0000100000
0000010000
0000000100
000000727200
000000007272
0000000010
,
1000000000
0100000000
0010000000
0001000000
0000010000
000072720000
0000000100
000000727200
0000000001
000000007272
,
72010000000
07201000000
72000000000
07200000000
0000001000
0000000100
0000000010
0000000001
0000100000
0000010000
,
44694469000000
440440000000
1565294000000
876933000000
000071530000
00005520000
000000007153
00000000552
000000715300
00000055200
,
15384370000000
53584030000000
3035835000000
33432015000000
000028310000
00003450000
000000283100
00000034500
000000002831
00000000345

G:=sub<GL(10,GF(73))| [0,1,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,0,0,72,1,0,0,0,0,0,0,0,0,72,0],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,72],[72,0,72,0,0,0,0,0,0,0,0,72,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0],[44,4,15,8,0,0,0,0,0,0,69,40,65,7,0,0,0,0,0,0,44,4,29,69,0,0,0,0,0,0,69,40,4,33,0,0,0,0,0,0,0,0,0,0,71,55,0,0,0,0,0,0,0,0,53,2,0,0,0,0,0,0,0,0,0,0,0,0,71,55,0,0,0,0,0,0,0,0,53,2,0,0,0,0,0,0,71,55,0,0,0,0,0,0,0,0,53,2,0,0],[15,53,30,33,0,0,0,0,0,0,38,58,3,43,0,0,0,0,0,0,43,40,58,20,0,0,0,0,0,0,70,30,35,15,0,0,0,0,0,0,0,0,0,0,28,3,0,0,0,0,0,0,0,0,31,45,0,0,0,0,0,0,0,0,0,0,28,3,0,0,0,0,0,0,0,0,31,45,0,0,0,0,0,0,0,0,0,0,28,3,0,0,0,0,0,0,0,0,31,45] >;
 

He3⋊3Q16 in GAP, Magma, Sage, TeX 

{\rm He}_3\rtimes_3Q_{16}
 
% in TeX
 
G:=Group("He3:3Q16");
 
// GroupNames label
 
G:=SmallGroup(432,86);
 
// by ID
 
G=gap.SmallGroup(432,86);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,85,92,254,58,571,4037,537,14118,7069]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^8=1,e^2=d^4,a*b=b*a,c*a*c^-1=a*b^-1,a*d=d*a,e*a*e^-1=a^-1,b*c=c*b,d*b*d^-1=e*b*e^-1=b^-1,d*c*d^-1=c^-1,c*e=e*c,e*d*e^-1=d^-1>;
 
// generators/relations
 

Export

Character table of He3⋊3Q16 in TeX

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