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## G = He3⋊3Q16order 432 = 24·33

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

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 He33Q16
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, C324C8, C3×Dic6, C324Q8, C32⋊C12, He33C4, C4×He3, C322Q16, C323Q16, He33C8, He33Q8, He34Q8, He33Q16
Quotients: C1, C2, C22, S3, D4, D6, Q16, D12, C3⋊D4, S32, Dic12, C3⋊Q16, C3⋊D12, C32⋊D6, C323Q16, He33D4, He33Q16

Character table of He33Q16

 class 1 2 3A 3B 3C 3D 4A 4B 4C 6A 6B 6C 6D 8A 8B 12A 12B 12C 12D 12E 12F 12G 12H 12I 12J 24A 24B 24C 24D size 1 1 2 6 6 12 2 36 36 2 6 6 12 18 18 4 6 6 12 12 12 36 36 36 36 18 18 18 18 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 1 1 -1 1 1 1 1 1 -1 -1 1 1 1 1 1 1 -1 1 1 -1 -1 -1 -1 -1 linear of order 2 ρ3 1 1 1 1 1 1 1 1 -1 1 1 1 1 -1 -1 1 1 1 1 1 1 1 -1 -1 1 -1 -1 -1 -1 linear of order 2 ρ4 1 1 1 1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 linear of order 2 ρ5 2 2 2 -1 2 -1 2 0 -2 2 -1 2 -1 0 0 2 2 2 -1 -1 -1 0 1 1 0 0 0 0 0 orthogonal lifted from D6 ρ6 2 2 2 2 2 2 -2 0 0 2 2 2 2 0 0 -2 -2 -2 -2 -2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ7 2 2 2 2 -1 -1 2 0 0 2 2 -1 -1 2 2 2 -1 -1 2 -1 -1 0 0 0 0 -1 -1 -1 -1 orthogonal lifted from S3 ρ8 2 2 2 2 -1 -1 2 0 0 2 2 -1 -1 -2 -2 2 -1 -1 2 -1 -1 0 0 0 0 1 1 1 1 orthogonal lifted from D6 ρ9 2 2 2 -1 2 -1 2 0 2 2 -1 2 -1 0 0 2 2 2 -1 -1 -1 0 -1 -1 0 0 0 0 0 orthogonal lifted from S3 ρ10 2 2 2 2 -1 -1 -2 0 0 2 2 -1 -1 0 0 -2 1 1 -2 1 1 0 0 0 0 √3 -√3 √3 -√3 orthogonal lifted from D12 ρ11 2 2 2 2 -1 -1 -2 0 0 2 2 -1 -1 0 0 -2 1 1 -2 1 1 0 0 0 0 -√3 √3 -√3 √3 orthogonal lifted from D12 ρ12 2 -2 2 2 2 2 0 0 0 -2 -2 -2 -2 √2 -√2 0 0 0 0 0 0 0 0 0 0 √2 √2 -√2 -√2 symplectic lifted from Q16, Schur index 2 ρ13 2 -2 2 2 2 2 0 0 0 -2 -2 -2 -2 -√2 √2 0 0 0 0 0 0 0 0 0 0 -√2 -√2 √2 √2 symplectic lifted from Q16, Schur index 2 ρ14 2 -2 2 2 -1 -1 0 0 0 -2 -2 1 1 -√2 √2 0 -√3 √3 0 √3 -√3 0 0 0 0 ζ83ζ3+ζ8ζ3+ζ8 ζ87ζ3+ζ87+ζ85ζ3 ζ83ζ32+ζ83+ζ8ζ32 ζ87ζ32+ζ85ζ32+ζ85 symplectic lifted from Dic12, Schur index 2 ρ15 2 -2 2 2 -1 -1 0 0 0 -2 -2 1 1 √2 -√2 0 -√3 √3 0 √3 -√3 0 0 0 0 ζ83ζ32+ζ83+ζ8ζ32 ζ87ζ32+ζ85ζ32+ζ85 ζ83ζ3+ζ8ζ3+ζ8 ζ87ζ3+ζ87+ζ85ζ3 symplectic lifted from Dic12, Schur index 2 ρ16 2 -2 2 2 -1 -1 0 0 0 -2 -2 1 1 -√2 √2 0 √3 -√3 0 -√3 √3 0 0 0 0 ζ87ζ3+ζ87+ζ85ζ3 ζ83ζ3+ζ8ζ3+ζ8 ζ87ζ32+ζ85ζ32+ζ85 ζ83ζ32+ζ83+ζ8ζ32 symplectic lifted from Dic12, Schur index 2 ρ17 2 -2 2 2 -1 -1 0 0 0 -2 -2 1 1 √2 -√2 0 √3 -√3 0 -√3 √3 0 0 0 0 ζ87ζ32+ζ85ζ32+ζ85 ζ83ζ32+ζ83+ζ8ζ32 ζ87ζ3+ζ87+ζ85ζ3 ζ83ζ3+ζ8ζ3+ζ8 symplectic lifted from Dic12, Schur index 2 ρ18 2 2 2 -1 2 -1 -2 0 0 2 -1 2 -1 0 0 -2 -2 -2 1 1 1 0 -√-3 √-3 0 0 0 0 0 complex lifted from C3⋊D4 ρ19 2 2 2 -1 2 -1 -2 0 0 2 -1 2 -1 0 0 -2 -2 -2 1 1 1 0 √-3 -√-3 0 0 0 0 0 complex lifted from C3⋊D4 ρ20 4 4 4 -2 -2 1 -4 0 0 4 -2 -2 1 0 0 -4 2 2 2 -1 -1 0 0 0 0 0 0 0 0 orthogonal lifted from C3⋊D12 ρ21 4 4 4 -2 -2 1 4 0 0 4 -2 -2 1 0 0 4 -2 -2 -2 1 1 0 0 0 0 0 0 0 0 orthogonal lifted from S32 ρ22 4 -4 4 -2 4 -2 0 0 0 -4 2 -4 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from C3⋊Q16, Schur index 2 ρ23 4 -4 4 -2 -2 1 0 0 0 -4 2 2 -1 0 0 0 2√3 -2√3 0 √3 -√3 0 0 0 0 0 0 0 0 symplectic lifted from C32⋊3Q16, Schur index 2 ρ24 4 -4 4 -2 -2 1 0 0 0 -4 2 2 -1 0 0 0 -2√3 2√3 0 -√3 √3 0 0 0 0 0 0 0 0 symplectic lifted from C32⋊3Q16, Schur index 2 ρ25 6 6 -3 0 0 0 6 2 0 -3 0 0 0 0 0 -3 0 0 0 0 0 -1 0 0 -1 0 0 0 0 orthogonal lifted from C32⋊D6 ρ26 6 6 -3 0 0 0 6 -2 0 -3 0 0 0 0 0 -3 0 0 0 0 0 1 0 0 1 0 0 0 0 orthogonal lifted from C32⋊D6 ρ27 6 6 -3 0 0 0 -6 0 0 -3 0 0 0 0 0 3 0 0 0 0 0 -√-3 0 0 √-3 0 0 0 0 complex lifted from He3⋊3D4 ρ28 6 6 -3 0 0 0 -6 0 0 -3 0 0 0 0 0 3 0 0 0 0 0 √-3 0 0 -√-3 0 0 0 0 complex lifted from He3⋊3D4 ρ29 12 -12 -6 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2

Smallest permutation representation of He33Q16
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 He33Q16 in GL10(𝔽73)

 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 1 0 0 0 0 0 0 0 0 0 0 1 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 72 72 0 0 0 0 0 0 0 0 1 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 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 0 1 0 0 0 0 0 0 0 0 72 72
,
 72 0 1 0 0 0 0 0 0 0 0 72 0 1 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 0 0 72 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 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
,
 44 69 44 69 0 0 0 0 0 0 4 40 4 40 0 0 0 0 0 0 15 65 29 4 0 0 0 0 0 0 8 7 69 33 0 0 0 0 0 0 0 0 0 0 71 53 0 0 0 0 0 0 0 0 55 2 0 0 0 0 0 0 0 0 0 0 0 0 71 53 0 0 0 0 0 0 0 0 55 2 0 0 0 0 0 0 71 53 0 0 0 0 0 0 0 0 55 2 0 0
,
 15 38 43 70 0 0 0 0 0 0 53 58 40 30 0 0 0 0 0 0 30 3 58 35 0 0 0 0 0 0 33 43 20 15 0 0 0 0 0 0 0 0 0 0 28 31 0 0 0 0 0 0 0 0 3 45 0 0 0 0 0 0 0 0 0 0 28 31 0 0 0 0 0 0 0 0 3 45 0 0 0 0 0 0 0 0 0 0 28 31 0 0 0 0 0 0 0 0 3 45

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] >;

He33Q16 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

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