metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.68D6, C3⋊C8⋊3Q8, C3⋊4(C8⋊Q8), C4⋊C4.73D6, C4.32(S3×Q8), (C2×C12).84D4, C6.28(C4⋊Q8), C12.32(C2×Q8), C42.C2.3S3, C12⋊2Q8.16C2, C6.Q16.13C2, C2.20(D4⋊D6), C6.121(C8⋊C22), (C2×C12).382C23, (C4×C12).112C22, C42.S3.5C2, C12.Q8.14C2, C2.8(Dic3⋊Q8), C2.21(Q8.14D6), C6.122(C8.C22), C4⋊Dic3.152C22, (C2×C6).513(C2×D4), (C2×C4).65(C3⋊D4), (C2×C3⋊C8).125C22, (C3×C42.C2).2C2, (C3×C4⋊C4).120C22, (C2×C4).480(C22×S3), C22.186(C2×C3⋊D4), SmallGroup(192,623)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.68D6
G = < a,b,c,d | a4=b4=1, c6=d2=a2, ab=ba, cac-1=a-1b2, dad-1=a-1, cbc-1=dbd-1=b-1, dcd-1=bc5 >
Subgroups: 224 in 90 conjugacy classes, 43 normal (27 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, Dic6, C2×Dic3, C2×C12, C2×C12, C8⋊C4, C4.Q8, C2.D8, C42.C2, C4⋊Q8, C2×C3⋊C8, C4⋊Dic3, C4⋊Dic3, C4×C12, C3×C4⋊C4, C3×C4⋊C4, C2×Dic6, C8⋊Q8, C42.S3, C6.Q16, C12.Q8, C12⋊2Q8, C3×C42.C2, C42.68D6
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, Dic3⋊Q8, D4⋊D6, Q8.14D6, C42.68D6
Character table of C42.68D6
| 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 | -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 | orthogonal lifted from D4⋊D6 |
| ρ24 | 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 |
| ρ25 | 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 | orthogonal lifted from D4⋊D6 |
| 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 | symplectic lifted from Q8.14D6, Schur index 2 |
| ρ28 | 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 |
| ρ29 | 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 | symplectic lifted from Q8.14D6, Schur index 2 |
| ρ30 | 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 |
►Regular action on 192 points
Generators in S192
(1 160 7 166)(2 121 8 127)(3 162 9 168)(4 123 10 129)(5 164 11 158)(6 125 12 131)(13 147 19 153)(14 27 20 33)(15 149 21 155)(16 29 22 35)(17 151 23 145)(18 31 24 25)(26 41 32 47)(28 43 34 37)(30 45 36 39)(38 150 44 156)(40 152 46 146)(42 154 48 148)(49 133 55 139)(50 91 56 85)(51 135 57 141)(52 93 58 87)(53 137 59 143)(54 95 60 89)(61 122 67 128)(62 163 68 157)(63 124 69 130)(64 165 70 159)(65 126 71 132)(66 167 72 161)(73 105 79 99)(74 184 80 190)(75 107 81 101)(76 186 82 192)(77 97 83 103)(78 188 84 182)(86 180 92 174)(88 170 94 176)(90 172 96 178)(98 114 104 120)(100 116 106 110)(102 118 108 112)(109 189 115 183)(111 191 117 185)(113 181 119 187)(134 173 140 179)(136 175 142 169)(138 177 144 171)
(1 60 65 171)(2 172 66 49)(3 50 67 173)(4 174 68 51)(5 52 69 175)(6 176 70 53)(7 54 71 177)(8 178 72 55)(9 56 61 179)(10 180 62 57)(11 58 63 169)(12 170 64 59)(13 84 41 120)(14 109 42 73)(15 74 43 110)(16 111 44 75)(17 76 45 112)(18 113 46 77)(19 78 47 114)(20 115 48 79)(21 80 37 116)(22 117 38 81)(23 82 39 118)(24 119 40 83)(25 187 152 103)(26 104 153 188)(27 189 154 105)(28 106 155 190)(29 191 156 107)(30 108 145 192)(31 181 146 97)(32 98 147 182)(33 183 148 99)(34 100 149 184)(35 185 150 101)(36 102 151 186)(85 122 134 168)(86 157 135 123)(87 124 136 158)(88 159 137 125)(89 126 138 160)(90 161 139 127)(91 128 140 162)(92 163 141 129)(93 130 142 164)(94 165 143 131)(95 132 144 166)(96 167 133 121)
(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)
(1 189 7 183)(2 147 8 153)(3 187 9 181)(4 145 10 151)(5 185 11 191)(6 155 12 149)(13 127 19 121)(14 95 20 89)(15 125 21 131)(16 93 22 87)(17 123 23 129)(18 91 24 85)(25 56 31 50)(26 66 32 72)(27 54 33 60)(28 64 34 70)(29 52 35 58)(30 62 36 68)(37 165 43 159)(38 136 44 142)(39 163 45 157)(40 134 46 140)(41 161 47 167)(42 144 48 138)(49 182 55 188)(51 192 57 186)(53 190 59 184)(61 97 67 103)(63 107 69 101)(65 105 71 99)(73 132 79 126)(74 137 80 143)(75 130 81 124)(76 135 82 141)(77 128 83 122)(78 133 84 139)(86 118 92 112)(88 116 94 110)(90 114 96 120)(98 178 104 172)(100 176 106 170)(102 174 108 180)(109 166 115 160)(111 164 117 158)(113 162 119 168)(146 173 152 179)(148 171 154 177)(150 169 156 175)
G:=sub<Sym(192)| (1,160,7,166)(2,121,8,127)(3,162,9,168)(4,123,10,129)(5,164,11,158)(6,125,12,131)(13,147,19,153)(14,27,20,33)(15,149,21,155)(16,29,22,35)(17,151,23,145)(18,31,24,25)(26,41,32,47)(28,43,34,37)(30,45,36,39)(38,150,44,156)(40,152,46,146)(42,154,48,148)(49,133,55,139)(50,91,56,85)(51,135,57,141)(52,93,58,87)(53,137,59,143)(54,95,60,89)(61,122,67,128)(62,163,68,157)(63,124,69,130)(64,165,70,159)(65,126,71,132)(66,167,72,161)(73,105,79,99)(74,184,80,190)(75,107,81,101)(76,186,82,192)(77,97,83,103)(78,188,84,182)(86,180,92,174)(88,170,94,176)(90,172,96,178)(98,114,104,120)(100,116,106,110)(102,118,108,112)(109,189,115,183)(111,191,117,185)(113,181,119,187)(134,173,140,179)(136,175,142,169)(138,177,144,171), (1,60,65,171)(2,172,66,49)(3,50,67,173)(4,174,68,51)(5,52,69,175)(6,176,70,53)(7,54,71,177)(8,178,72,55)(9,56,61,179)(10,180,62,57)(11,58,63,169)(12,170,64,59)(13,84,41,120)(14,109,42,73)(15,74,43,110)(16,111,44,75)(17,76,45,112)(18,113,46,77)(19,78,47,114)(20,115,48,79)(21,80,37,116)(22,117,38,81)(23,82,39,118)(24,119,40,83)(25,187,152,103)(26,104,153,188)(27,189,154,105)(28,106,155,190)(29,191,156,107)(30,108,145,192)(31,181,146,97)(32,98,147,182)(33,183,148,99)(34,100,149,184)(35,185,150,101)(36,102,151,186)(85,122,134,168)(86,157,135,123)(87,124,136,158)(88,159,137,125)(89,126,138,160)(90,161,139,127)(91,128,140,162)(92,163,141,129)(93,130,142,164)(94,165,143,131)(95,132,144,166)(96,167,133,121), (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), (1,189,7,183)(2,147,8,153)(3,187,9,181)(4,145,10,151)(5,185,11,191)(6,155,12,149)(13,127,19,121)(14,95,20,89)(15,125,21,131)(16,93,22,87)(17,123,23,129)(18,91,24,85)(25,56,31,50)(26,66,32,72)(27,54,33,60)(28,64,34,70)(29,52,35,58)(30,62,36,68)(37,165,43,159)(38,136,44,142)(39,163,45,157)(40,134,46,140)(41,161,47,167)(42,144,48,138)(49,182,55,188)(51,192,57,186)(53,190,59,184)(61,97,67,103)(63,107,69,101)(65,105,71,99)(73,132,79,126)(74,137,80,143)(75,130,81,124)(76,135,82,141)(77,128,83,122)(78,133,84,139)(86,118,92,112)(88,116,94,110)(90,114,96,120)(98,178,104,172)(100,176,106,170)(102,174,108,180)(109,166,115,160)(111,164,117,158)(113,162,119,168)(146,173,152,179)(148,171,154,177)(150,169,156,175)>;
G:=Group( (1,160,7,166)(2,121,8,127)(3,162,9,168)(4,123,10,129)(5,164,11,158)(6,125,12,131)(13,147,19,153)(14,27,20,33)(15,149,21,155)(16,29,22,35)(17,151,23,145)(18,31,24,25)(26,41,32,47)(28,43,34,37)(30,45,36,39)(38,150,44,156)(40,152,46,146)(42,154,48,148)(49,133,55,139)(50,91,56,85)(51,135,57,141)(52,93,58,87)(53,137,59,143)(54,95,60,89)(61,122,67,128)(62,163,68,157)(63,124,69,130)(64,165,70,159)(65,126,71,132)(66,167,72,161)(73,105,79,99)(74,184,80,190)(75,107,81,101)(76,186,82,192)(77,97,83,103)(78,188,84,182)(86,180,92,174)(88,170,94,176)(90,172,96,178)(98,114,104,120)(100,116,106,110)(102,118,108,112)(109,189,115,183)(111,191,117,185)(113,181,119,187)(134,173,140,179)(136,175,142,169)(138,177,144,171), (1,60,65,171)(2,172,66,49)(3,50,67,173)(4,174,68,51)(5,52,69,175)(6,176,70,53)(7,54,71,177)(8,178,72,55)(9,56,61,179)(10,180,62,57)(11,58,63,169)(12,170,64,59)(13,84,41,120)(14,109,42,73)(15,74,43,110)(16,111,44,75)(17,76,45,112)(18,113,46,77)(19,78,47,114)(20,115,48,79)(21,80,37,116)(22,117,38,81)(23,82,39,118)(24,119,40,83)(25,187,152,103)(26,104,153,188)(27,189,154,105)(28,106,155,190)(29,191,156,107)(30,108,145,192)(31,181,146,97)(32,98,147,182)(33,183,148,99)(34,100,149,184)(35,185,150,101)(36,102,151,186)(85,122,134,168)(86,157,135,123)(87,124,136,158)(88,159,137,125)(89,126,138,160)(90,161,139,127)(91,128,140,162)(92,163,141,129)(93,130,142,164)(94,165,143,131)(95,132,144,166)(96,167,133,121), (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), (1,189,7,183)(2,147,8,153)(3,187,9,181)(4,145,10,151)(5,185,11,191)(6,155,12,149)(13,127,19,121)(14,95,20,89)(15,125,21,131)(16,93,22,87)(17,123,23,129)(18,91,24,85)(25,56,31,50)(26,66,32,72)(27,54,33,60)(28,64,34,70)(29,52,35,58)(30,62,36,68)(37,165,43,159)(38,136,44,142)(39,163,45,157)(40,134,46,140)(41,161,47,167)(42,144,48,138)(49,182,55,188)(51,192,57,186)(53,190,59,184)(61,97,67,103)(63,107,69,101)(65,105,71,99)(73,132,79,126)(74,137,80,143)(75,130,81,124)(76,135,82,141)(77,128,83,122)(78,133,84,139)(86,118,92,112)(88,116,94,110)(90,114,96,120)(98,178,104,172)(100,176,106,170)(102,174,108,180)(109,166,115,160)(111,164,117,158)(113,162,119,168)(146,173,152,179)(148,171,154,177)(150,169,156,175) );
G=PermutationGroup([[(1,160,7,166),(2,121,8,127),(3,162,9,168),(4,123,10,129),(5,164,11,158),(6,125,12,131),(13,147,19,153),(14,27,20,33),(15,149,21,155),(16,29,22,35),(17,151,23,145),(18,31,24,25),(26,41,32,47),(28,43,34,37),(30,45,36,39),(38,150,44,156),(40,152,46,146),(42,154,48,148),(49,133,55,139),(50,91,56,85),(51,135,57,141),(52,93,58,87),(53,137,59,143),(54,95,60,89),(61,122,67,128),(62,163,68,157),(63,124,69,130),(64,165,70,159),(65,126,71,132),(66,167,72,161),(73,105,79,99),(74,184,80,190),(75,107,81,101),(76,186,82,192),(77,97,83,103),(78,188,84,182),(86,180,92,174),(88,170,94,176),(90,172,96,178),(98,114,104,120),(100,116,106,110),(102,118,108,112),(109,189,115,183),(111,191,117,185),(113,181,119,187),(134,173,140,179),(136,175,142,169),(138,177,144,171)], [(1,60,65,171),(2,172,66,49),(3,50,67,173),(4,174,68,51),(5,52,69,175),(6,176,70,53),(7,54,71,177),(8,178,72,55),(9,56,61,179),(10,180,62,57),(11,58,63,169),(12,170,64,59),(13,84,41,120),(14,109,42,73),(15,74,43,110),(16,111,44,75),(17,76,45,112),(18,113,46,77),(19,78,47,114),(20,115,48,79),(21,80,37,116),(22,117,38,81),(23,82,39,118),(24,119,40,83),(25,187,152,103),(26,104,153,188),(27,189,154,105),(28,106,155,190),(29,191,156,107),(30,108,145,192),(31,181,146,97),(32,98,147,182),(33,183,148,99),(34,100,149,184),(35,185,150,101),(36,102,151,186),(85,122,134,168),(86,157,135,123),(87,124,136,158),(88,159,137,125),(89,126,138,160),(90,161,139,127),(91,128,140,162),(92,163,141,129),(93,130,142,164),(94,165,143,131),(95,132,144,166),(96,167,133,121)], [(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)], [(1,189,7,183),(2,147,8,153),(3,187,9,181),(4,145,10,151),(5,185,11,191),(6,155,12,149),(13,127,19,121),(14,95,20,89),(15,125,21,131),(16,93,22,87),(17,123,23,129),(18,91,24,85),(25,56,31,50),(26,66,32,72),(27,54,33,60),(28,64,34,70),(29,52,35,58),(30,62,36,68),(37,165,43,159),(38,136,44,142),(39,163,45,157),(40,134,46,140),(41,161,47,167),(42,144,48,138),(49,182,55,188),(51,192,57,186),(53,190,59,184),(61,97,67,103),(63,107,69,101),(65,105,71,99),(73,132,79,126),(74,137,80,143),(75,130,81,124),(76,135,82,141),(77,128,83,122),(78,133,84,139),(86,118,92,112),(88,116,94,110),(90,114,96,120),(98,178,104,172),(100,176,106,170),(102,174,108,180),(109,166,115,160),(111,164,117,158),(113,162,119,168),(146,173,152,179),(148,171,154,177),(150,169,156,175)]])
Matrix representation of C42.68D6 ►in GL6(𝔽73)
| 72 | 16 | 0 | 0 | 0 | 0 |
| 9 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 14 | 0 | 0 |
| 0 | 0 | 59 | 66 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 14 |
| 0 | 0 | 0 | 0 | 59 | 66 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 71 | 0 |
| 0 | 0 | 0 | 72 | 0 | 71 |
| 0 | 0 | 1 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 | 0 | 1 |
| 1 | 2 | 0 | 0 | 0 | 0 |
| 72 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 30 | 33 | 21 |
| 0 | 0 | 43 | 51 | 52 | 12 |
| 0 | 0 | 45 | 17 | 65 | 43 |
| 0 | 0 | 56 | 28 | 30 | 22 |
| 72 | 71 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 34 | 6 | 58 |
| 0 | 0 | 33 | 72 | 52 | 67 |
| 0 | 0 | 2 | 68 | 72 | 39 |
| 0 | 0 | 66 | 71 | 40 | 1 |
G:=sub<GL(6,GF(73))| [72,9,0,0,0,0,16,1,0,0,0,0,0,0,7,59,0,0,0,0,14,66,0,0,0,0,0,0,7,59,0,0,0,0,14,66],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,1,0,0,0,0,72,0,1,0,0,71,0,1,0,0,0,0,71,0,1],[1,72,0,0,0,0,2,72,0,0,0,0,0,0,8,43,45,56,0,0,30,51,17,28,0,0,33,52,65,30,0,0,21,12,43,22],[72,1,0,0,0,0,71,1,0,0,0,0,0,0,1,33,2,66,0,0,34,72,68,71,0,0,6,52,72,40,0,0,58,67,39,1] >;
C42.68D6 in GAP, Magma, Sage, TeX
C_4^2._{68}D_6 % in TeX
G:=Group("C4^2.68D6"); // GroupNames label
G:=SmallGroup(192,623);
// by ID
G=gap.SmallGroup(192,623);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,477,64,422,471,58,438,102,6278]);
// 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*b^2,d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b*c^5>;
// generators/relations
Export