metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.76D6, C3⋊C8⋊4Q8, C3⋊5(C8⋊Q8), C4⋊Q8.7S3, C4⋊C4.81D6, C4.36(S3×Q8), C6.32(C4⋊Q8), C12.37(C2×Q8), (C2×C12).293D4, C6.97(C8⋊C22), C6.Q16.16C2, C12.6Q8.8C2, (C2×C12).399C23, (C4×C12).128C22, C6.93(C8.C22), C42.S3.7C2, C12.Q8.17C2, C2.18(D12⋊6C22), C4⋊Dic3.159C22, C2.12(Dic3⋊Q8), C2.14(Q8.11D6), (C3×C4⋊Q8).7C2, (C2×C6).530(C2×D4), (C2×C4).71(C3⋊D4), (C2×C3⋊C8).133C22, (C3×C4⋊C4).128C22, (C2×C4).496(C22×S3), C22.202(C2×C3⋊D4), SmallGroup(192,640)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.76D6
G = < a,b,c,d | a4=b4=1, c6=d2=a2, ab=ba, cac-1=a-1, dad-1=a-1b2, cbc-1=dbd-1=b-1, dcd-1=bc5 >
Subgroups: 208 in 90 conjugacy classes, 43 normal (25 characteristic)
C1, C2, C3, C4, C4, C22, C6, C8, C2×C4, C2×C4, Q8, Dic3, C12, C12, C2×C6, C42, C4⋊C4, C4⋊C4, C2×C8, C2×Q8, C3⋊C8, C2×Dic3, C2×C12, C2×C12, C3×Q8, C8⋊C4, C4.Q8, C2.D8, C42.C2, C4⋊Q8, C2×C3⋊C8, Dic3⋊C4, C4⋊Dic3, C4×C12, C3×C4⋊C4, C3×C4⋊C4, C6×Q8, C8⋊Q8, C42.S3, C6.Q16, C12.Q8, C12.6Q8, C3×C4⋊Q8, C42.76D6
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2×D4, C2×Q8, C3⋊D4, C22×S3, C4⋊Q8, C8⋊C22, C8.C22, S3×Q8, C2×C3⋊D4, C8⋊Q8, D12⋊6C22, Q8.11D6, Dic3⋊Q8, C42.76D6
Character table of C42.76D6
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | 12J | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 24 | 24 | 2 | 2 | 2 | 12 | 12 | 12 | 12 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | |
| 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 | 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 | -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 | -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 | 1 | linear of order 2 |
| ρ5 | 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 | 1 | linear of order 2 |
| ρ6 | 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 | -1 | linear of order 2 |
| ρ7 | 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 | 1 | linear of order 2 |
| ρ8 | 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 | -1 | linear of order 2 |
| ρ9 | 2 | 2 | 2 | 2 | 2 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | 2 | -1 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ11 | 2 | 2 | 2 | 2 | -1 | 2 | 2 | -2 | -2 | -2 | 2 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | orthogonal lifted from D6 |
| ρ12 | 2 | 2 | 2 | 2 | -1 | 2 | 2 | -2 | -2 | 2 | -2 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | orthogonal lifted from D6 |
| ρ13 | 2 | 2 | 2 | 2 | -1 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ14 | 2 | 2 | 2 | 2 | 2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ15 | 2 | 2 | -2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | -2 | 2 | 0 | 0 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ16 | 2 | 2 | -2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | -2 | 0 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ17 | 2 | 2 | -2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 2 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ18 | 2 | 2 | -2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 2 | 0 | 0 | -2 | 0 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ19 | 2 | 2 | 2 | 2 | -1 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | -1 | -1 | 1 | √-3 | -√-3 | -√-3 | √-3 | complex lifted from C3⋊D4 |
| ρ20 | 2 | 2 | 2 | 2 | -1 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | 1 | 1 | 1 | √-3 | √-3 | -√-3 | -√-3 | complex lifted from C3⋊D4 |
| ρ21 | 2 | 2 | 2 | 2 | -1 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | 1 | 1 | 1 | -√-3 | -√-3 | √-3 | √-3 | complex lifted from C3⋊D4 |
| ρ22 | 2 | 2 | 2 | 2 | -1 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | -1 | -1 | 1 | -√-3 | √-3 | √-3 | -√-3 | complex lifted from C3⋊D4 |
| ρ23 | 4 | -4 | 4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C8⋊C22 |
| ρ24 | 4 | 4 | -4 | -4 | -2 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | symplectic lifted from S3×Q8, Schur index 2 |
| ρ25 | 4 | -4 | -4 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C8.C22, Schur index 2 |
| ρ26 | 4 | 4 | -4 | -4 | -2 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | symplectic lifted from S3×Q8, Schur index 2 |
| ρ27 | 4 | -4 | 4 | -4 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 2√-3 | -2√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from D12⋊6C22 |
| ρ28 | 4 | -4 | 4 | -4 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | -2√-3 | 2√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from D12⋊6C22 |
| ρ29 | 4 | -4 | -4 | 4 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2√-3 | -2√-3 | 0 | 0 | 0 | 0 | 0 | complex lifted from Q8.11D6 |
| ρ30 | 4 | -4 | -4 | 4 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2√-3 | 2√-3 | 0 | 0 | 0 | 0 | 0 | complex lifted from Q8.11D6 |
►Regular action on 192 points
Generators in S192
(1 145 7 151)(2 152 8 146)(3 147 9 153)(4 154 10 148)(5 149 11 155)(6 156 12 150)(13 65 19 71)(14 72 20 66)(15 67 21 61)(16 62 22 68)(17 69 23 63)(18 64 24 70)(25 138 31 144)(26 133 32 139)(27 140 33 134)(28 135 34 141)(29 142 35 136)(30 137 36 143)(37 186 43 192)(38 181 44 187)(39 188 45 182)(40 183 46 189)(41 190 47 184)(42 185 48 191)(49 112 55 118)(50 119 56 113)(51 114 57 120)(52 109 58 115)(53 116 59 110)(54 111 60 117)(73 93 79 87)(74 88 80 94)(75 95 81 89)(76 90 82 96)(77 85 83 91)(78 92 84 86)(97 122 103 128)(98 129 104 123)(99 124 105 130)(100 131 106 125)(101 126 107 132)(102 121 108 127)(157 175 163 169)(158 170 164 176)(159 177 165 171)(160 172 166 178)(161 179 167 173)(162 174 168 180)
(1 30 84 39)(2 40 73 31)(3 32 74 41)(4 42 75 33)(5 34 76 43)(6 44 77 35)(7 36 78 45)(8 46 79 25)(9 26 80 47)(10 48 81 27)(11 28 82 37)(12 38 83 29)(13 121 55 163)(14 164 56 122)(15 123 57 165)(16 166 58 124)(17 125 59 167)(18 168 60 126)(19 127 49 157)(20 158 50 128)(21 129 51 159)(22 160 52 130)(23 131 53 161)(24 162 54 132)(61 104 114 177)(62 178 115 105)(63 106 116 179)(64 180 117 107)(65 108 118 169)(66 170 119 97)(67 98 120 171)(68 172 109 99)(69 100 110 173)(70 174 111 101)(71 102 112 175)(72 176 113 103)(85 136 156 187)(86 188 145 137)(87 138 146 189)(88 190 147 139)(89 140 148 191)(90 192 149 141)(91 142 150 181)(92 182 151 143)(93 144 152 183)(94 184 153 133)(95 134 154 185)(96 186 155 135)
(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)(145 146 147 148 149 150 151 152 153 154 155 156)(157 158 159 160 161 162 163 164 165 166 167 168)(169 170 171 172 173 174 175 176 177 178 179 180)(181 182 183 184 185 186 187 188 189 190 191 192)
(1 108 7 102)(2 111 8 117)(3 106 9 100)(4 109 10 115)(5 104 11 98)(6 119 12 113)(13 182 19 188)(14 156 20 150)(15 192 21 186)(16 154 22 148)(17 190 23 184)(18 152 24 146)(25 107 31 101)(26 69 32 63)(27 105 33 99)(28 67 34 61)(29 103 35 97)(30 65 36 71)(37 120 43 114)(38 176 44 170)(39 118 45 112)(40 174 46 180)(41 116 47 110)(42 172 48 178)(49 137 55 143)(50 91 56 85)(51 135 57 141)(52 89 58 95)(53 133 59 139)(54 87 60 93)(62 75 68 81)(64 73 70 79)(66 83 72 77)(74 179 80 173)(76 177 82 171)(78 175 84 169)(86 121 92 127)(88 131 94 125)(90 129 96 123)(122 187 128 181)(124 185 130 191)(126 183 132 189)(134 160 140 166)(136 158 142 164)(138 168 144 162)(145 163 151 157)(147 161 153 167)(149 159 155 165)
G:=sub<Sym(192)| (1,145,7,151)(2,152,8,146)(3,147,9,153)(4,154,10,148)(5,149,11,155)(6,156,12,150)(13,65,19,71)(14,72,20,66)(15,67,21,61)(16,62,22,68)(17,69,23,63)(18,64,24,70)(25,138,31,144)(26,133,32,139)(27,140,33,134)(28,135,34,141)(29,142,35,136)(30,137,36,143)(37,186,43,192)(38,181,44,187)(39,188,45,182)(40,183,46,189)(41,190,47,184)(42,185,48,191)(49,112,55,118)(50,119,56,113)(51,114,57,120)(52,109,58,115)(53,116,59,110)(54,111,60,117)(73,93,79,87)(74,88,80,94)(75,95,81,89)(76,90,82,96)(77,85,83,91)(78,92,84,86)(97,122,103,128)(98,129,104,123)(99,124,105,130)(100,131,106,125)(101,126,107,132)(102,121,108,127)(157,175,163,169)(158,170,164,176)(159,177,165,171)(160,172,166,178)(161,179,167,173)(162,174,168,180), (1,30,84,39)(2,40,73,31)(3,32,74,41)(4,42,75,33)(5,34,76,43)(6,44,77,35)(7,36,78,45)(8,46,79,25)(9,26,80,47)(10,48,81,27)(11,28,82,37)(12,38,83,29)(13,121,55,163)(14,164,56,122)(15,123,57,165)(16,166,58,124)(17,125,59,167)(18,168,60,126)(19,127,49,157)(20,158,50,128)(21,129,51,159)(22,160,52,130)(23,131,53,161)(24,162,54,132)(61,104,114,177)(62,178,115,105)(63,106,116,179)(64,180,117,107)(65,108,118,169)(66,170,119,97)(67,98,120,171)(68,172,109,99)(69,100,110,173)(70,174,111,101)(71,102,112,175)(72,176,113,103)(85,136,156,187)(86,188,145,137)(87,138,146,189)(88,190,147,139)(89,140,148,191)(90,192,149,141)(91,142,150,181)(92,182,151,143)(93,144,152,183)(94,184,153,133)(95,134,154,185)(96,186,155,135), (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)(145,146,147,148,149,150,151,152,153,154,155,156)(157,158,159,160,161,162,163,164,165,166,167,168)(169,170,171,172,173,174,175,176,177,178,179,180)(181,182,183,184,185,186,187,188,189,190,191,192), (1,108,7,102)(2,111,8,117)(3,106,9,100)(4,109,10,115)(5,104,11,98)(6,119,12,113)(13,182,19,188)(14,156,20,150)(15,192,21,186)(16,154,22,148)(17,190,23,184)(18,152,24,146)(25,107,31,101)(26,69,32,63)(27,105,33,99)(28,67,34,61)(29,103,35,97)(30,65,36,71)(37,120,43,114)(38,176,44,170)(39,118,45,112)(40,174,46,180)(41,116,47,110)(42,172,48,178)(49,137,55,143)(50,91,56,85)(51,135,57,141)(52,89,58,95)(53,133,59,139)(54,87,60,93)(62,75,68,81)(64,73,70,79)(66,83,72,77)(74,179,80,173)(76,177,82,171)(78,175,84,169)(86,121,92,127)(88,131,94,125)(90,129,96,123)(122,187,128,181)(124,185,130,191)(126,183,132,189)(134,160,140,166)(136,158,142,164)(138,168,144,162)(145,163,151,157)(147,161,153,167)(149,159,155,165)>;
G:=Group( (1,145,7,151)(2,152,8,146)(3,147,9,153)(4,154,10,148)(5,149,11,155)(6,156,12,150)(13,65,19,71)(14,72,20,66)(15,67,21,61)(16,62,22,68)(17,69,23,63)(18,64,24,70)(25,138,31,144)(26,133,32,139)(27,140,33,134)(28,135,34,141)(29,142,35,136)(30,137,36,143)(37,186,43,192)(38,181,44,187)(39,188,45,182)(40,183,46,189)(41,190,47,184)(42,185,48,191)(49,112,55,118)(50,119,56,113)(51,114,57,120)(52,109,58,115)(53,116,59,110)(54,111,60,117)(73,93,79,87)(74,88,80,94)(75,95,81,89)(76,90,82,96)(77,85,83,91)(78,92,84,86)(97,122,103,128)(98,129,104,123)(99,124,105,130)(100,131,106,125)(101,126,107,132)(102,121,108,127)(157,175,163,169)(158,170,164,176)(159,177,165,171)(160,172,166,178)(161,179,167,173)(162,174,168,180), (1,30,84,39)(2,40,73,31)(3,32,74,41)(4,42,75,33)(5,34,76,43)(6,44,77,35)(7,36,78,45)(8,46,79,25)(9,26,80,47)(10,48,81,27)(11,28,82,37)(12,38,83,29)(13,121,55,163)(14,164,56,122)(15,123,57,165)(16,166,58,124)(17,125,59,167)(18,168,60,126)(19,127,49,157)(20,158,50,128)(21,129,51,159)(22,160,52,130)(23,131,53,161)(24,162,54,132)(61,104,114,177)(62,178,115,105)(63,106,116,179)(64,180,117,107)(65,108,118,169)(66,170,119,97)(67,98,120,171)(68,172,109,99)(69,100,110,173)(70,174,111,101)(71,102,112,175)(72,176,113,103)(85,136,156,187)(86,188,145,137)(87,138,146,189)(88,190,147,139)(89,140,148,191)(90,192,149,141)(91,142,150,181)(92,182,151,143)(93,144,152,183)(94,184,153,133)(95,134,154,185)(96,186,155,135), (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)(145,146,147,148,149,150,151,152,153,154,155,156)(157,158,159,160,161,162,163,164,165,166,167,168)(169,170,171,172,173,174,175,176,177,178,179,180)(181,182,183,184,185,186,187,188,189,190,191,192), (1,108,7,102)(2,111,8,117)(3,106,9,100)(4,109,10,115)(5,104,11,98)(6,119,12,113)(13,182,19,188)(14,156,20,150)(15,192,21,186)(16,154,22,148)(17,190,23,184)(18,152,24,146)(25,107,31,101)(26,69,32,63)(27,105,33,99)(28,67,34,61)(29,103,35,97)(30,65,36,71)(37,120,43,114)(38,176,44,170)(39,118,45,112)(40,174,46,180)(41,116,47,110)(42,172,48,178)(49,137,55,143)(50,91,56,85)(51,135,57,141)(52,89,58,95)(53,133,59,139)(54,87,60,93)(62,75,68,81)(64,73,70,79)(66,83,72,77)(74,179,80,173)(76,177,82,171)(78,175,84,169)(86,121,92,127)(88,131,94,125)(90,129,96,123)(122,187,128,181)(124,185,130,191)(126,183,132,189)(134,160,140,166)(136,158,142,164)(138,168,144,162)(145,163,151,157)(147,161,153,167)(149,159,155,165) );
G=PermutationGroup([[(1,145,7,151),(2,152,8,146),(3,147,9,153),(4,154,10,148),(5,149,11,155),(6,156,12,150),(13,65,19,71),(14,72,20,66),(15,67,21,61),(16,62,22,68),(17,69,23,63),(18,64,24,70),(25,138,31,144),(26,133,32,139),(27,140,33,134),(28,135,34,141),(29,142,35,136),(30,137,36,143),(37,186,43,192),(38,181,44,187),(39,188,45,182),(40,183,46,189),(41,190,47,184),(42,185,48,191),(49,112,55,118),(50,119,56,113),(51,114,57,120),(52,109,58,115),(53,116,59,110),(54,111,60,117),(73,93,79,87),(74,88,80,94),(75,95,81,89),(76,90,82,96),(77,85,83,91),(78,92,84,86),(97,122,103,128),(98,129,104,123),(99,124,105,130),(100,131,106,125),(101,126,107,132),(102,121,108,127),(157,175,163,169),(158,170,164,176),(159,177,165,171),(160,172,166,178),(161,179,167,173),(162,174,168,180)], [(1,30,84,39),(2,40,73,31),(3,32,74,41),(4,42,75,33),(5,34,76,43),(6,44,77,35),(7,36,78,45),(8,46,79,25),(9,26,80,47),(10,48,81,27),(11,28,82,37),(12,38,83,29),(13,121,55,163),(14,164,56,122),(15,123,57,165),(16,166,58,124),(17,125,59,167),(18,168,60,126),(19,127,49,157),(20,158,50,128),(21,129,51,159),(22,160,52,130),(23,131,53,161),(24,162,54,132),(61,104,114,177),(62,178,115,105),(63,106,116,179),(64,180,117,107),(65,108,118,169),(66,170,119,97),(67,98,120,171),(68,172,109,99),(69,100,110,173),(70,174,111,101),(71,102,112,175),(72,176,113,103),(85,136,156,187),(86,188,145,137),(87,138,146,189),(88,190,147,139),(89,140,148,191),(90,192,149,141),(91,142,150,181),(92,182,151,143),(93,144,152,183),(94,184,153,133),(95,134,154,185),(96,186,155,135)], [(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),(145,146,147,148,149,150,151,152,153,154,155,156),(157,158,159,160,161,162,163,164,165,166,167,168),(169,170,171,172,173,174,175,176,177,178,179,180),(181,182,183,184,185,186,187,188,189,190,191,192)], [(1,108,7,102),(2,111,8,117),(3,106,9,100),(4,109,10,115),(5,104,11,98),(6,119,12,113),(13,182,19,188),(14,156,20,150),(15,192,21,186),(16,154,22,148),(17,190,23,184),(18,152,24,146),(25,107,31,101),(26,69,32,63),(27,105,33,99),(28,67,34,61),(29,103,35,97),(30,65,36,71),(37,120,43,114),(38,176,44,170),(39,118,45,112),(40,174,46,180),(41,116,47,110),(42,172,48,178),(49,137,55,143),(50,91,56,85),(51,135,57,141),(52,89,58,95),(53,133,59,139),(54,87,60,93),(62,75,68,81),(64,73,70,79),(66,83,72,77),(74,179,80,173),(76,177,82,171),(78,175,84,169),(86,121,92,127),(88,131,94,125),(90,129,96,123),(122,187,128,181),(124,185,130,191),(126,183,132,189),(134,160,140,166),(136,158,142,164),(138,168,144,162),(145,163,151,157),(147,161,153,167),(149,159,155,165)]])
Matrix representation of C42.76D6 ►in GL8(𝔽73)
| 17 | 21 | 2 | 62 | 0 | 0 | 0 | 0 |
| 52 | 38 | 11 | 64 | 0 | 0 | 0 | 0 |
| 57 | 10 | 56 | 52 | 0 | 0 | 0 | 0 |
| 63 | 67 | 21 | 35 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 61 | 47 | 58 | 69 |
| 0 | 0 | 0 | 0 | 0 | 25 | 22 | 45 |
| 0 | 0 | 0 | 0 | 58 | 23 | 12 | 0 |
| 0 | 0 | 0 | 0 | 56 | 30 | 3 | 48 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 48 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 13 | 72 | 2 |
| 0 | 0 | 0 | 0 | 17 | 13 | 72 | 1 |
| 0 | 1 | 0 | 2 | 0 | 0 | 0 | 0 |
| 72 | 1 | 71 | 2 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 72 | 0 | 0 | 0 | 0 |
| 1 | 72 | 1 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 56 | 0 | 3 |
| 0 | 0 | 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 25 | 0 | 17 |
| 51 | 56 | 70 | 47 | 0 | 0 | 0 | 0 |
| 34 | 22 | 44 | 3 | 0 | 0 | 0 | 0 |
| 38 | 13 | 54 | 9 | 0 | 0 | 0 | 0 |
| 51 | 35 | 63 | 19 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 45 | 46 | 68 | 5 |
| 0 | 0 | 0 | 0 | 68 | 28 | 29 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 24 | 21 |
| 0 | 0 | 0 | 0 | 35 | 1 | 65 | 49 |
G:=sub<GL(8,GF(73))| [17,52,57,63,0,0,0,0,21,38,10,67,0,0,0,0,2,11,56,21,0,0,0,0,62,64,52,35,0,0,0,0,0,0,0,0,61,0,58,56,0,0,0,0,47,25,23,30,0,0,0,0,58,22,12,3,0,0,0,0,69,45,0,48],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,3,0,17,0,0,0,0,48,72,13,13,0,0,0,0,0,0,72,72,0,0,0,0,0,0,2,1],[0,72,0,1,0,0,0,0,1,1,72,72,0,0,0,0,0,71,0,1,0,0,0,0,2,2,72,72,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,56,0,25,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,17],[51,34,38,51,0,0,0,0,56,22,13,35,0,0,0,0,70,44,54,63,0,0,0,0,47,3,9,19,0,0,0,0,0,0,0,0,45,68,0,35,0,0,0,0,46,28,1,1,0,0,0,0,68,29,24,65,0,0,0,0,5,0,21,49] >;
C42.76D6 in GAP, Magma, Sage, TeX
C_4^2._{76}D_6 % in TeX
G:=Group("C4^2.76D6"); // GroupNames label
G:=SmallGroup(192,640);
// by ID
G=gap.SmallGroup(192,640);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,477,64,422,135,58,438,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^6=d^2=a^2,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a^-1*b^2,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b*c^5>;
// generators/relations
Export