non-abelian, supersoluble, monomial
Aliases: He3⋊2Q16, C12.83S32, (C3×C12).3D6, (C2×He3).8D4, He3⋊4C8.2C2, He3⋊3Q8.1C2, C6.3(D6⋊S3), C32⋊4Q8.1S3, C2.5(He3⋊2D4), C4.10(C32⋊D6), C32⋊1(C3⋊Q16), (C4×He3).3C22, C3.1(C32⋊2Q16), (C3×C6).3(C3⋊D4), SmallGroup(432,80)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for He3⋊2Q16
G = < a,b,c,d,e | a3=b3=c3=d8=1, e2=d4, ab=ba, cac-1=ab-1, dad-1=a-1, ae=ea, bc=cb, bd=db, ebe-1=b-1, dcd-1=ece-1=c-1, ede-1=d-1 >
Subgroups: 371 in 71 conjugacy classes, 21 normal (11 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, C3×Dic6, C32⋊4Q8, C32⋊C12, C4×He3, C32⋊3Q16, He3⋊4C8, He3⋊3Q8, He3⋊2Q16
Quotients: C1, C2, C22, S3, D4, D6, Q16, C3⋊D4, S32, C3⋊Q16, D6⋊S3, C32⋊D6, C32⋊2Q16, He3⋊2D4, He3⋊2Q16
Character table of He3⋊2Q16
| 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 | 2 | 2 | 12 | 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 | 2 | 0 | 2 | 2 | -1 | -1 | 0 | 0 | 2 | 2 | -1 | -1 | -1 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3 |
| ρ6 | 2 | 2 | 2 | 2 | -1 | -1 | 2 | 0 | 2 | 2 | -1 | 2 | -1 | 0 | 0 | 2 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | -1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from S3 |
| ρ7 | 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 |
| ρ8 | 2 | 2 | 2 | -1 | 2 | -1 | 2 | -2 | 0 | 2 | 2 | -1 | -1 | 0 | 0 | 2 | 2 | -1 | -1 | -1 | 2 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D6 |
| ρ9 | 2 | 2 | 2 | 2 | -1 | -1 | 2 | 0 | -2 | 2 | -1 | 2 | -1 | 0 | 0 | 2 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | orthogonal lifted from D6 |
| ρ10 | 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 |
| ρ11 | 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 |
| ρ12 | 2 | 2 | 2 | 2 | -1 | -1 | -2 | 0 | 0 | 2 | -1 | 2 | -1 | 0 | 0 | -2 | -2 | -2 | 1 | 1 | 1 | 0 | 0 | -√-3 | √-3 | 0 | 0 | 0 | 0 | complex lifted from C3⋊D4 |
| ρ13 | 2 | 2 | 2 | -1 | 2 | -1 | -2 | 0 | 0 | 2 | 2 | -1 | -1 | 0 | 0 | -2 | -2 | 1 | 1 | 1 | -2 | √-3 | -√-3 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3⋊D4 |
| ρ14 | 2 | 2 | 2 | 2 | -1 | -1 | -2 | 0 | 0 | 2 | -1 | 2 | -1 | 0 | 0 | -2 | -2 | -2 | 1 | 1 | 1 | 0 | 0 | √-3 | -√-3 | 0 | 0 | 0 | 0 | complex lifted from C3⋊D4 |
| ρ15 | 2 | 2 | 2 | -1 | 2 | -1 | -2 | 0 | 0 | 2 | 2 | -1 | -1 | 0 | 0 | -2 | -2 | 1 | 1 | 1 | -2 | -√-3 | √-3 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3⋊D4 |
| ρ16 | 4 | 4 | 4 | -2 | -2 | 1 | 4 | 0 | 0 | 4 | -2 | -2 | 1 | 0 | 0 | 4 | 4 | -2 | 1 | 1 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S32 |
| ρ17 | 4 | -4 | 4 | -2 | 4 | -2 | 0 | 0 | 0 | -4 | -4 | 2 | 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 |
| ρ18 | 4 | -4 | 4 | 4 | -2 | -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 |
| ρ19 | 4 | 4 | 4 | -2 | -2 | 1 | -4 | 0 | 0 | 4 | -2 | -2 | 1 | 0 | 0 | -4 | -4 | 2 | -1 | -1 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from D6⋊S3, Schur index 2 |
| ρ20 | 4 | -4 | 4 | -2 | -2 | 1 | 0 | 0 | 0 | -4 | 2 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | -3i | 3i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C32⋊2Q16 |
| ρ21 | 4 | -4 | 4 | -2 | -2 | 1 | 0 | 0 | 0 | -4 | 2 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 3i | -3i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C32⋊2Q16 |
| ρ22 | 6 | 6 | -3 | 0 | 0 | 0 | 6 | 0 | 0 | -3 | 0 | 0 | 0 | 2 | 2 | -3 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from C32⋊D6 |
| ρ23 | 6 | 6 | -3 | 0 | 0 | 0 | 6 | 0 | 0 | -3 | 0 | 0 | 0 | -2 | -2 | -3 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | orthogonal lifted from C32⋊D6 |
| ρ24 | 6 | 6 | -3 | 0 | 0 | 0 | -6 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √3 | -√3 | -√3 | √3 | orthogonal lifted from He3⋊2D4 |
| ρ25 | 6 | 6 | -3 | 0 | 0 | 0 | -6 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√3 | √3 | √3 | -√3 | orthogonal lifted from He3⋊2D4 |
| ρ26 | 6 | -6 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 3 | 0 | 0 | 0 | -√2 | √2 | 3√3 | -3√3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ87ζ3+ζ87+ζ85ζ3 | ζ83ζ32+ζ83+ζ8ζ32 | ζ83ζ3+ζ8ζ3+ζ8 | ζ87ζ32+ζ85ζ32+ζ85 | symplectic faithful, Schur index 2 |
| ρ27 | 6 | -6 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 3 | 0 | 0 | 0 | √2 | -√2 | 3√3 | -3√3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ87ζ32+ζ85ζ32+ζ85 | ζ83ζ3+ζ8ζ3+ζ8 | ζ83ζ32+ζ83+ζ8ζ32 | ζ87ζ3+ζ87+ζ85ζ3 | symplectic faithful, Schur index 2 |
| ρ28 | 6 | -6 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 3 | 0 | 0 | 0 | √2 | -√2 | -3√3 | 3√3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ83ζ32+ζ83+ζ8ζ32 | ζ87ζ3+ζ87+ζ85ζ3 | ζ87ζ32+ζ85ζ32+ζ85 | ζ83ζ3+ζ8ζ3+ζ8 | symplectic faithful, Schur index 2 |
| ρ29 | 6 | -6 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 3 | 0 | 0 | 0 | -√2 | √2 | -3√3 | 3√3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ83ζ3+ζ8ζ3+ζ8 | ζ87ζ32+ζ85ζ32+ζ85 | ζ87ζ3+ζ87+ζ85ζ3 | ζ83ζ32+ζ83+ζ8ζ32 | symplectic faithful, Schur index 2 |
►On 144 points
Generators in S144
(17 65 78)(18 79 66)(19 67 80)(20 73 68)(21 69 74)(22 75 70)(23 71 76)(24 77 72)(25 89 110)(26 111 90)(27 91 112)(28 105 92)(29 93 106)(30 107 94)(31 95 108)(32 109 96)(33 102 137)(34 138 103)(35 104 139)(36 140 97)(37 98 141)(38 142 99)(39 100 143)(40 144 101)(49 122 133)(50 134 123)(51 124 135)(52 136 125)(53 126 129)(54 130 127)(55 128 131)(56 132 121)
(1 15 117)(2 16 118)(3 9 119)(4 10 120)(5 11 113)(6 12 114)(7 13 115)(8 14 116)(17 78 65)(18 79 66)(19 80 67)(20 73 68)(21 74 69)(22 75 70)(23 76 71)(24 77 72)(25 110 89)(26 111 90)(27 112 91)(28 105 92)(29 106 93)(30 107 94)(31 108 95)(32 109 96)(33 102 137)(34 103 138)(35 104 139)(36 97 140)(37 98 141)(38 99 142)(39 100 143)(40 101 144)(41 61 84)(42 62 85)(43 63 86)(44 64 87)(45 57 88)(46 58 81)(47 59 82)(48 60 83)(49 133 122)(50 134 123)(51 135 124)(52 136 125)(53 129 126)(54 130 127)(55 131 128)(56 132 121)
(1 133 22)(2 23 134)(3 135 24)(4 17 136)(5 129 18)(6 19 130)(7 131 20)(8 21 132)(9 124 77)(10 78 125)(11 126 79)(12 80 127)(13 128 73)(14 74 121)(15 122 75)(16 76 123)(25 35 82)(26 83 36)(27 37 84)(28 85 38)(29 39 86)(30 87 40)(31 33 88)(32 81 34)(41 112 98)(42 99 105)(43 106 100)(44 101 107)(45 108 102)(46 103 109)(47 110 104)(48 97 111)(49 70 117)(50 118 71)(51 72 119)(52 120 65)(53 66 113)(54 114 67)(55 68 115)(56 116 69)(57 95 137)(58 138 96)(59 89 139)(60 140 90)(61 91 141)(62 142 92)(63 93 143)(64 144 94)
(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 61 5 57)(2 60 6 64)(3 59 7 63)(4 58 8 62)(9 47 13 43)(10 46 14 42)(11 45 15 41)(12 44 16 48)(17 96 21 92)(18 95 22 91)(19 94 23 90)(20 93 24 89)(25 68 29 72)(26 67 30 71)(27 66 31 70)(28 65 32 69)(33 49 37 53)(34 56 38 52)(35 55 39 51)(36 54 40 50)(73 106 77 110)(74 105 78 109)(75 112 79 108)(76 111 80 107)(81 116 85 120)(82 115 86 119)(83 114 87 118)(84 113 88 117)(97 127 101 123)(98 126 102 122)(99 125 103 121)(100 124 104 128)(129 137 133 141)(130 144 134 140)(131 143 135 139)(132 142 136 138)
G:=sub<Sym(144)| (17,65,78)(18,79,66)(19,67,80)(20,73,68)(21,69,74)(22,75,70)(23,71,76)(24,77,72)(25,89,110)(26,111,90)(27,91,112)(28,105,92)(29,93,106)(30,107,94)(31,95,108)(32,109,96)(33,102,137)(34,138,103)(35,104,139)(36,140,97)(37,98,141)(38,142,99)(39,100,143)(40,144,101)(49,122,133)(50,134,123)(51,124,135)(52,136,125)(53,126,129)(54,130,127)(55,128,131)(56,132,121), (1,15,117)(2,16,118)(3,9,119)(4,10,120)(5,11,113)(6,12,114)(7,13,115)(8,14,116)(17,78,65)(18,79,66)(19,80,67)(20,73,68)(21,74,69)(22,75,70)(23,76,71)(24,77,72)(25,110,89)(26,111,90)(27,112,91)(28,105,92)(29,106,93)(30,107,94)(31,108,95)(32,109,96)(33,102,137)(34,103,138)(35,104,139)(36,97,140)(37,98,141)(38,99,142)(39,100,143)(40,101,144)(41,61,84)(42,62,85)(43,63,86)(44,64,87)(45,57,88)(46,58,81)(47,59,82)(48,60,83)(49,133,122)(50,134,123)(51,135,124)(52,136,125)(53,129,126)(54,130,127)(55,131,128)(56,132,121), (1,133,22)(2,23,134)(3,135,24)(4,17,136)(5,129,18)(6,19,130)(7,131,20)(8,21,132)(9,124,77)(10,78,125)(11,126,79)(12,80,127)(13,128,73)(14,74,121)(15,122,75)(16,76,123)(25,35,82)(26,83,36)(27,37,84)(28,85,38)(29,39,86)(30,87,40)(31,33,88)(32,81,34)(41,112,98)(42,99,105)(43,106,100)(44,101,107)(45,108,102)(46,103,109)(47,110,104)(48,97,111)(49,70,117)(50,118,71)(51,72,119)(52,120,65)(53,66,113)(54,114,67)(55,68,115)(56,116,69)(57,95,137)(58,138,96)(59,89,139)(60,140,90)(61,91,141)(62,142,92)(63,93,143)(64,144,94), (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), (1,61,5,57)(2,60,6,64)(3,59,7,63)(4,58,8,62)(9,47,13,43)(10,46,14,42)(11,45,15,41)(12,44,16,48)(17,96,21,92)(18,95,22,91)(19,94,23,90)(20,93,24,89)(25,68,29,72)(26,67,30,71)(27,66,31,70)(28,65,32,69)(33,49,37,53)(34,56,38,52)(35,55,39,51)(36,54,40,50)(73,106,77,110)(74,105,78,109)(75,112,79,108)(76,111,80,107)(81,116,85,120)(82,115,86,119)(83,114,87,118)(84,113,88,117)(97,127,101,123)(98,126,102,122)(99,125,103,121)(100,124,104,128)(129,137,133,141)(130,144,134,140)(131,143,135,139)(132,142,136,138)>;
G:=Group( (17,65,78)(18,79,66)(19,67,80)(20,73,68)(21,69,74)(22,75,70)(23,71,76)(24,77,72)(25,89,110)(26,111,90)(27,91,112)(28,105,92)(29,93,106)(30,107,94)(31,95,108)(32,109,96)(33,102,137)(34,138,103)(35,104,139)(36,140,97)(37,98,141)(38,142,99)(39,100,143)(40,144,101)(49,122,133)(50,134,123)(51,124,135)(52,136,125)(53,126,129)(54,130,127)(55,128,131)(56,132,121), (1,15,117)(2,16,118)(3,9,119)(4,10,120)(5,11,113)(6,12,114)(7,13,115)(8,14,116)(17,78,65)(18,79,66)(19,80,67)(20,73,68)(21,74,69)(22,75,70)(23,76,71)(24,77,72)(25,110,89)(26,111,90)(27,112,91)(28,105,92)(29,106,93)(30,107,94)(31,108,95)(32,109,96)(33,102,137)(34,103,138)(35,104,139)(36,97,140)(37,98,141)(38,99,142)(39,100,143)(40,101,144)(41,61,84)(42,62,85)(43,63,86)(44,64,87)(45,57,88)(46,58,81)(47,59,82)(48,60,83)(49,133,122)(50,134,123)(51,135,124)(52,136,125)(53,129,126)(54,130,127)(55,131,128)(56,132,121), (1,133,22)(2,23,134)(3,135,24)(4,17,136)(5,129,18)(6,19,130)(7,131,20)(8,21,132)(9,124,77)(10,78,125)(11,126,79)(12,80,127)(13,128,73)(14,74,121)(15,122,75)(16,76,123)(25,35,82)(26,83,36)(27,37,84)(28,85,38)(29,39,86)(30,87,40)(31,33,88)(32,81,34)(41,112,98)(42,99,105)(43,106,100)(44,101,107)(45,108,102)(46,103,109)(47,110,104)(48,97,111)(49,70,117)(50,118,71)(51,72,119)(52,120,65)(53,66,113)(54,114,67)(55,68,115)(56,116,69)(57,95,137)(58,138,96)(59,89,139)(60,140,90)(61,91,141)(62,142,92)(63,93,143)(64,144,94), (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), (1,61,5,57)(2,60,6,64)(3,59,7,63)(4,58,8,62)(9,47,13,43)(10,46,14,42)(11,45,15,41)(12,44,16,48)(17,96,21,92)(18,95,22,91)(19,94,23,90)(20,93,24,89)(25,68,29,72)(26,67,30,71)(27,66,31,70)(28,65,32,69)(33,49,37,53)(34,56,38,52)(35,55,39,51)(36,54,40,50)(73,106,77,110)(74,105,78,109)(75,112,79,108)(76,111,80,107)(81,116,85,120)(82,115,86,119)(83,114,87,118)(84,113,88,117)(97,127,101,123)(98,126,102,122)(99,125,103,121)(100,124,104,128)(129,137,133,141)(130,144,134,140)(131,143,135,139)(132,142,136,138) );
G=PermutationGroup([[(17,65,78),(18,79,66),(19,67,80),(20,73,68),(21,69,74),(22,75,70),(23,71,76),(24,77,72),(25,89,110),(26,111,90),(27,91,112),(28,105,92),(29,93,106),(30,107,94),(31,95,108),(32,109,96),(33,102,137),(34,138,103),(35,104,139),(36,140,97),(37,98,141),(38,142,99),(39,100,143),(40,144,101),(49,122,133),(50,134,123),(51,124,135),(52,136,125),(53,126,129),(54,130,127),(55,128,131),(56,132,121)], [(1,15,117),(2,16,118),(3,9,119),(4,10,120),(5,11,113),(6,12,114),(7,13,115),(8,14,116),(17,78,65),(18,79,66),(19,80,67),(20,73,68),(21,74,69),(22,75,70),(23,76,71),(24,77,72),(25,110,89),(26,111,90),(27,112,91),(28,105,92),(29,106,93),(30,107,94),(31,108,95),(32,109,96),(33,102,137),(34,103,138),(35,104,139),(36,97,140),(37,98,141),(38,99,142),(39,100,143),(40,101,144),(41,61,84),(42,62,85),(43,63,86),(44,64,87),(45,57,88),(46,58,81),(47,59,82),(48,60,83),(49,133,122),(50,134,123),(51,135,124),(52,136,125),(53,129,126),(54,130,127),(55,131,128),(56,132,121)], [(1,133,22),(2,23,134),(3,135,24),(4,17,136),(5,129,18),(6,19,130),(7,131,20),(8,21,132),(9,124,77),(10,78,125),(11,126,79),(12,80,127),(13,128,73),(14,74,121),(15,122,75),(16,76,123),(25,35,82),(26,83,36),(27,37,84),(28,85,38),(29,39,86),(30,87,40),(31,33,88),(32,81,34),(41,112,98),(42,99,105),(43,106,100),(44,101,107),(45,108,102),(46,103,109),(47,110,104),(48,97,111),(49,70,117),(50,118,71),(51,72,119),(52,120,65),(53,66,113),(54,114,67),(55,68,115),(56,116,69),(57,95,137),(58,138,96),(59,89,139),(60,140,90),(61,91,141),(62,142,92),(63,93,143),(64,144,94)], [(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)], [(1,61,5,57),(2,60,6,64),(3,59,7,63),(4,58,8,62),(9,47,13,43),(10,46,14,42),(11,45,15,41),(12,44,16,48),(17,96,21,92),(18,95,22,91),(19,94,23,90),(20,93,24,89),(25,68,29,72),(26,67,30,71),(27,66,31,70),(28,65,32,69),(33,49,37,53),(34,56,38,52),(35,55,39,51),(36,54,40,50),(73,106,77,110),(74,105,78,109),(75,112,79,108),(76,111,80,107),(81,116,85,120),(82,115,86,119),(83,114,87,118),(84,113,88,117),(97,127,101,123),(98,126,102,122),(99,125,103,121),(100,124,104,128),(129,137,133,141),(130,144,134,140),(131,143,135,139),(132,142,136,138)]])
Matrix representation of He3⋊2Q16 ►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 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 37 | 72 | 0 | 72 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 36 | 1 | 72 | 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 |
| 12 | 43 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 55 | 61 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 10 | 39 | 32 | 48 | 0 | 0 | 0 | 0 | 0 | 0 |
| 49 | 63 | 7 | 41 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 14 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 59 | 66 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 7 | 14 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 59 | 66 |
| 0 | 0 | 0 | 0 | 0 | 0 | 7 | 14 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 59 | 66 | 0 | 0 |
| 70 | 45 | 67 | 17 | 0 | 0 | 0 | 0 | 0 | 0 |
| 28 | 42 | 56 | 11 | 0 | 0 | 0 | 0 | 0 | 0 |
| 17 | 0 | 3 | 28 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 17 | 45 | 31 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 44 | 25 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 54 | 29 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 44 | 25 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 54 | 29 |
| 0 | 0 | 0 | 0 | 0 | 0 | 44 | 25 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 54 | 29 | 0 | 0 |
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],[72,72,37,1,0,0,0,0,0,0,1,0,72,36,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0],[12,55,10,49,0,0,0,0,0,0,43,61,39,63,0,0,0,0,0,0,0,0,32,7,0,0,0,0,0,0,0,0,48,41,0,0,0,0,0,0,0,0,0,0,7,59,0,0,0,0,0,0,0,0,14,66,0,0,0,0,0,0,0,0,0,0,0,0,7,59,0,0,0,0,0,0,0,0,14,66,0,0,0,0,0,0,7,59,0,0,0,0,0,0,0,0,14,66,0,0],[70,28,17,0,0,0,0,0,0,0,45,42,0,17,0,0,0,0,0,0,67,56,3,45,0,0,0,0,0,0,17,11,28,31,0,0,0,0,0,0,0,0,0,0,44,54,0,0,0,0,0,0,0,0,25,29,0,0,0,0,0,0,0,0,0,0,0,0,44,54,0,0,0,0,0,0,0,0,25,29,0,0,0,0,0,0,44,54,0,0,0,0,0,0,0,0,25,29,0,0] >;
He3⋊2Q16 in GAP, Magma, Sage, TeX
{\rm He}_3\rtimes_2Q_{16} % in TeX
G:=Group("He3:2Q16"); // GroupNames label
G:=SmallGroup(432,80);
// by ID
G=gap.SmallGroup(432,80);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,85,64,254,135,58,571,4037,537,14118,7069]);
// 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,d*a*d^-1=a^-1,a*e=e*a,b*c=c*b,b*d=d*b,e*b*e^-1=b^-1,d*c*d^-1=e*c*e^-1=c^-1,e*d*e^-1=d^-1>;
// generators/relations
Export