metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.71D6, C4⋊C4.78D6, (C2×C12).86D4, C42.C2.5S3, C12.73(C4○D4), C12⋊2Q8.17C2, (C2×C12).387C23, (C4×C12).117C22, C4.15(Q8⋊3S3), C6.SD16.13C2, C6.57(C4.4D4), C42.S3.6C2, C2.22(Q8.14D6), C6.123(C8.C22), C2.10(C12.23D4), C3⋊3(C42.30C22), (C2×Dic6).110C22, (C2×C6).518(C2×D4), (C2×C4).68(C3⋊D4), (C2×C3⋊C8).128C22, (C3×C42.C2).4C2, (C3×C4⋊C4).125C22, (C2×C4).485(C22×S3), C22.191(C2×C3⋊D4), SmallGroup(192,628)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.71D6
G = < a,b,c,d | a4=b4=1, c6=a2, d2=a2b, ab=ba, cac-1=a-1b2, dad-1=ab2, cbc-1=b-1, bd=db, dcd-1=b-1c5 >
Subgroups: 224 in 90 conjugacy classes, 39 normal (15 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C6, C8, C2×C4, 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, C2×C12, C8⋊C4, Q8⋊C4, C42.C2, C4⋊Q8, C2×C3⋊C8, C4⋊Dic3, C4×C12, C3×C4⋊C4, C3×C4⋊C4, C2×Dic6, C42.30C22, C42.S3, C6.SD16, C12⋊2Q8, C3×C42.C2, C42.71D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C3⋊D4, C22×S3, C4.4D4, C8.C22, Q8⋊3S3, C2×C3⋊D4, C42.30C22, C12.23D4, Q8.14D6, C42.71D6
Character table of C42.71D6
| 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 | 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 |
| 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 | -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 |
| ρ15 | 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 |
| ρ16 | 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 |
| ρ17 | 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 |
| ρ18 | 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 |
| ρ19 | 2 | -2 | 2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 0 | 2i | -2i | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ20 | 2 | -2 | 2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 0 | -2i | 2i | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ21 | 2 | -2 | 2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 2i | 0 | 0 | -2i | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ22 | 2 | -2 | 2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | -2i | 0 | 0 | 2i | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ23 | 4 | -4 | 4 | -4 | -2 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from Q8⋊3S3, Schur index 2 |
| ρ24 | 4 | -4 | 4 | -4 | -2 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from Q8⋊3S3, 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 | 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 |
| ρ27 | 4 | 4 | -4 | -4 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | -2√3 | 0 | 0 | 2√3 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8.14D6, Schur index 2 |
| ρ28 | 4 | 4 | -4 | -4 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 2√3 | 0 | 0 | -2√3 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8.14D6, Schur index 2 |
| ρ29 | 4 | -4 | -4 | 4 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2√3 | -2√3 | 0 | 0 | 0 | 0 | symplectic lifted from Q8.14D6, Schur index 2 |
| ρ30 | 4 | -4 | -4 | 4 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2√3 | 2√3 | 0 | 0 | 0 | 0 | symplectic lifted from Q8.14D6, Schur index 2 |
►Regular action on 192 points
Generators in S192
(1 137 7 143)(2 50 8 56)(3 139 9 133)(4 52 10 58)(5 141 11 135)(6 54 12 60)(13 166 19 160)(14 89 20 95)(15 168 21 162)(16 91 22 85)(17 158 23 164)(18 93 24 87)(25 86 31 92)(26 165 32 159)(27 88 33 94)(28 167 34 161)(29 90 35 96)(30 157 36 163)(37 51 43 57)(38 140 44 134)(39 53 45 59)(40 142 46 136)(41 55 47 49)(42 144 48 138)(61 190 67 184)(62 150 68 156)(63 192 69 186)(64 152 70 146)(65 182 71 188)(66 154 72 148)(73 115 79 109)(74 123 80 129)(75 117 81 111)(76 125 82 131)(77 119 83 113)(78 127 84 121)(97 183 103 189)(98 155 104 149)(99 185 105 191)(100 145 106 151)(101 187 107 181)(102 147 108 153)(110 175 116 169)(112 177 118 171)(114 179 120 173)(122 180 128 174)(124 170 130 176)(126 172 132 178)
(1 21 41 29)(2 30 42 22)(3 23 43 31)(4 32 44 24)(5 13 45 33)(6 34 46 14)(7 15 47 35)(8 36 48 16)(9 17 37 25)(10 26 38 18)(11 19 39 27)(12 28 40 20)(49 96 143 168)(50 157 144 85)(51 86 133 158)(52 159 134 87)(53 88 135 160)(54 161 136 89)(55 90 137 162)(56 163 138 91)(57 92 139 164)(58 165 140 93)(59 94 141 166)(60 167 142 95)(61 73 98 180)(62 169 99 74)(63 75 100 170)(64 171 101 76)(65 77 102 172)(66 173 103 78)(67 79 104 174)(68 175 105 80)(69 81 106 176)(70 177 107 82)(71 83 108 178)(72 179 97 84)(109 149 122 184)(110 185 123 150)(111 151 124 186)(112 187 125 152)(113 153 126 188)(114 189 127 154)(115 155 128 190)(116 191 129 156)(117 145 130 192)(118 181 131 146)(119 147 132 182)(120 183 121 148)
(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)
(1 145 15 124 41 192 35 111)(2 123 36 156 42 110 16 191)(3 155 17 122 43 190 25 109)(4 121 26 154 44 120 18 189)(5 153 19 132 45 188 27 119)(6 131 28 152 46 118 20 187)(7 151 21 130 47 186 29 117)(8 129 30 150 48 116 22 185)(9 149 23 128 37 184 31 115)(10 127 32 148 38 114 24 183)(11 147 13 126 39 182 33 113)(12 125 34 146 40 112 14 181)(49 100 90 176 143 63 162 81)(50 175 163 99 144 80 91 62)(51 98 92 174 133 61 164 79)(52 173 165 97 134 78 93 72)(53 108 94 172 135 71 166 77)(54 171 167 107 136 76 95 70)(55 106 96 170 137 69 168 75)(56 169 157 105 138 74 85 68)(57 104 86 180 139 67 158 73)(58 179 159 103 140 84 87 66)(59 102 88 178 141 65 160 83)(60 177 161 101 142 82 89 64)
G:=sub<Sym(192)| (1,137,7,143)(2,50,8,56)(3,139,9,133)(4,52,10,58)(5,141,11,135)(6,54,12,60)(13,166,19,160)(14,89,20,95)(15,168,21,162)(16,91,22,85)(17,158,23,164)(18,93,24,87)(25,86,31,92)(26,165,32,159)(27,88,33,94)(28,167,34,161)(29,90,35,96)(30,157,36,163)(37,51,43,57)(38,140,44,134)(39,53,45,59)(40,142,46,136)(41,55,47,49)(42,144,48,138)(61,190,67,184)(62,150,68,156)(63,192,69,186)(64,152,70,146)(65,182,71,188)(66,154,72,148)(73,115,79,109)(74,123,80,129)(75,117,81,111)(76,125,82,131)(77,119,83,113)(78,127,84,121)(97,183,103,189)(98,155,104,149)(99,185,105,191)(100,145,106,151)(101,187,107,181)(102,147,108,153)(110,175,116,169)(112,177,118,171)(114,179,120,173)(122,180,128,174)(124,170,130,176)(126,172,132,178), (1,21,41,29)(2,30,42,22)(3,23,43,31)(4,32,44,24)(5,13,45,33)(6,34,46,14)(7,15,47,35)(8,36,48,16)(9,17,37,25)(10,26,38,18)(11,19,39,27)(12,28,40,20)(49,96,143,168)(50,157,144,85)(51,86,133,158)(52,159,134,87)(53,88,135,160)(54,161,136,89)(55,90,137,162)(56,163,138,91)(57,92,139,164)(58,165,140,93)(59,94,141,166)(60,167,142,95)(61,73,98,180)(62,169,99,74)(63,75,100,170)(64,171,101,76)(65,77,102,172)(66,173,103,78)(67,79,104,174)(68,175,105,80)(69,81,106,176)(70,177,107,82)(71,83,108,178)(72,179,97,84)(109,149,122,184)(110,185,123,150)(111,151,124,186)(112,187,125,152)(113,153,126,188)(114,189,127,154)(115,155,128,190)(116,191,129,156)(117,145,130,192)(118,181,131,146)(119,147,132,182)(120,183,121,148), (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), (1,145,15,124,41,192,35,111)(2,123,36,156,42,110,16,191)(3,155,17,122,43,190,25,109)(4,121,26,154,44,120,18,189)(5,153,19,132,45,188,27,119)(6,131,28,152,46,118,20,187)(7,151,21,130,47,186,29,117)(8,129,30,150,48,116,22,185)(9,149,23,128,37,184,31,115)(10,127,32,148,38,114,24,183)(11,147,13,126,39,182,33,113)(12,125,34,146,40,112,14,181)(49,100,90,176,143,63,162,81)(50,175,163,99,144,80,91,62)(51,98,92,174,133,61,164,79)(52,173,165,97,134,78,93,72)(53,108,94,172,135,71,166,77)(54,171,167,107,136,76,95,70)(55,106,96,170,137,69,168,75)(56,169,157,105,138,74,85,68)(57,104,86,180,139,67,158,73)(58,179,159,103,140,84,87,66)(59,102,88,178,141,65,160,83)(60,177,161,101,142,82,89,64)>;
G:=Group( (1,137,7,143)(2,50,8,56)(3,139,9,133)(4,52,10,58)(5,141,11,135)(6,54,12,60)(13,166,19,160)(14,89,20,95)(15,168,21,162)(16,91,22,85)(17,158,23,164)(18,93,24,87)(25,86,31,92)(26,165,32,159)(27,88,33,94)(28,167,34,161)(29,90,35,96)(30,157,36,163)(37,51,43,57)(38,140,44,134)(39,53,45,59)(40,142,46,136)(41,55,47,49)(42,144,48,138)(61,190,67,184)(62,150,68,156)(63,192,69,186)(64,152,70,146)(65,182,71,188)(66,154,72,148)(73,115,79,109)(74,123,80,129)(75,117,81,111)(76,125,82,131)(77,119,83,113)(78,127,84,121)(97,183,103,189)(98,155,104,149)(99,185,105,191)(100,145,106,151)(101,187,107,181)(102,147,108,153)(110,175,116,169)(112,177,118,171)(114,179,120,173)(122,180,128,174)(124,170,130,176)(126,172,132,178), (1,21,41,29)(2,30,42,22)(3,23,43,31)(4,32,44,24)(5,13,45,33)(6,34,46,14)(7,15,47,35)(8,36,48,16)(9,17,37,25)(10,26,38,18)(11,19,39,27)(12,28,40,20)(49,96,143,168)(50,157,144,85)(51,86,133,158)(52,159,134,87)(53,88,135,160)(54,161,136,89)(55,90,137,162)(56,163,138,91)(57,92,139,164)(58,165,140,93)(59,94,141,166)(60,167,142,95)(61,73,98,180)(62,169,99,74)(63,75,100,170)(64,171,101,76)(65,77,102,172)(66,173,103,78)(67,79,104,174)(68,175,105,80)(69,81,106,176)(70,177,107,82)(71,83,108,178)(72,179,97,84)(109,149,122,184)(110,185,123,150)(111,151,124,186)(112,187,125,152)(113,153,126,188)(114,189,127,154)(115,155,128,190)(116,191,129,156)(117,145,130,192)(118,181,131,146)(119,147,132,182)(120,183,121,148), (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), (1,145,15,124,41,192,35,111)(2,123,36,156,42,110,16,191)(3,155,17,122,43,190,25,109)(4,121,26,154,44,120,18,189)(5,153,19,132,45,188,27,119)(6,131,28,152,46,118,20,187)(7,151,21,130,47,186,29,117)(8,129,30,150,48,116,22,185)(9,149,23,128,37,184,31,115)(10,127,32,148,38,114,24,183)(11,147,13,126,39,182,33,113)(12,125,34,146,40,112,14,181)(49,100,90,176,143,63,162,81)(50,175,163,99,144,80,91,62)(51,98,92,174,133,61,164,79)(52,173,165,97,134,78,93,72)(53,108,94,172,135,71,166,77)(54,171,167,107,136,76,95,70)(55,106,96,170,137,69,168,75)(56,169,157,105,138,74,85,68)(57,104,86,180,139,67,158,73)(58,179,159,103,140,84,87,66)(59,102,88,178,141,65,160,83)(60,177,161,101,142,82,89,64) );
G=PermutationGroup([[(1,137,7,143),(2,50,8,56),(3,139,9,133),(4,52,10,58),(5,141,11,135),(6,54,12,60),(13,166,19,160),(14,89,20,95),(15,168,21,162),(16,91,22,85),(17,158,23,164),(18,93,24,87),(25,86,31,92),(26,165,32,159),(27,88,33,94),(28,167,34,161),(29,90,35,96),(30,157,36,163),(37,51,43,57),(38,140,44,134),(39,53,45,59),(40,142,46,136),(41,55,47,49),(42,144,48,138),(61,190,67,184),(62,150,68,156),(63,192,69,186),(64,152,70,146),(65,182,71,188),(66,154,72,148),(73,115,79,109),(74,123,80,129),(75,117,81,111),(76,125,82,131),(77,119,83,113),(78,127,84,121),(97,183,103,189),(98,155,104,149),(99,185,105,191),(100,145,106,151),(101,187,107,181),(102,147,108,153),(110,175,116,169),(112,177,118,171),(114,179,120,173),(122,180,128,174),(124,170,130,176),(126,172,132,178)], [(1,21,41,29),(2,30,42,22),(3,23,43,31),(4,32,44,24),(5,13,45,33),(6,34,46,14),(7,15,47,35),(8,36,48,16),(9,17,37,25),(10,26,38,18),(11,19,39,27),(12,28,40,20),(49,96,143,168),(50,157,144,85),(51,86,133,158),(52,159,134,87),(53,88,135,160),(54,161,136,89),(55,90,137,162),(56,163,138,91),(57,92,139,164),(58,165,140,93),(59,94,141,166),(60,167,142,95),(61,73,98,180),(62,169,99,74),(63,75,100,170),(64,171,101,76),(65,77,102,172),(66,173,103,78),(67,79,104,174),(68,175,105,80),(69,81,106,176),(70,177,107,82),(71,83,108,178),(72,179,97,84),(109,149,122,184),(110,185,123,150),(111,151,124,186),(112,187,125,152),(113,153,126,188),(114,189,127,154),(115,155,128,190),(116,191,129,156),(117,145,130,192),(118,181,131,146),(119,147,132,182),(120,183,121,148)], [(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)], [(1,145,15,124,41,192,35,111),(2,123,36,156,42,110,16,191),(3,155,17,122,43,190,25,109),(4,121,26,154,44,120,18,189),(5,153,19,132,45,188,27,119),(6,131,28,152,46,118,20,187),(7,151,21,130,47,186,29,117),(8,129,30,150,48,116,22,185),(9,149,23,128,37,184,31,115),(10,127,32,148,38,114,24,183),(11,147,13,126,39,182,33,113),(12,125,34,146,40,112,14,181),(49,100,90,176,143,63,162,81),(50,175,163,99,144,80,91,62),(51,98,92,174,133,61,164,79),(52,173,165,97,134,78,93,72),(53,108,94,172,135,71,166,77),(54,171,167,107,136,76,95,70),(55,106,96,170,137,69,168,75),(56,169,157,105,138,74,85,68),(57,104,86,180,139,67,158,73),(58,179,159,103,140,84,87,66),(59,102,88,178,141,65,160,83),(60,177,161,101,142,82,89,64)]])
Matrix representation of C42.71D6 ►in GL8(𝔽73)
| 30 | 11 | 0 | 0 | 0 | 0 | 0 | 0 |
| 11 | 43 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 | 16 | 69 |
| 0 | 0 | 0 | 0 | 0 | 28 | 69 | 57 |
| 0 | 0 | 0 | 0 | 57 | 4 | 45 | 0 |
| 0 | 0 | 0 | 0 | 4 | 16 | 0 | 45 |
| 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 | 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 | 1 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 46 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 46 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 29 | 54 | 0 | 0 | 0 | 0 |
| 0 | 0 | 25 | 44 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 58 | 58 | 35 | 63 |
| 0 | 0 | 0 | 0 | 15 | 58 | 63 | 38 |
| 0 | 0 | 0 | 0 | 63 | 38 | 15 | 58 |
| 0 | 0 | 0 | 0 | 38 | 10 | 15 | 15 |
G:=sub<GL(8,GF(73))| [30,11,0,0,0,0,0,0,11,43,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,28,0,57,4,0,0,0,0,0,28,4,16,0,0,0,0,16,69,45,0,0,0,0,0,69,57,0,45],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0],[0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[46,0,0,0,0,0,0,0,0,46,0,0,0,0,0,0,0,0,29,25,0,0,0,0,0,0,54,44,0,0,0,0,0,0,0,0,58,15,63,38,0,0,0,0,58,58,38,10,0,0,0,0,35,63,15,15,0,0,0,0,63,38,58,15] >;
C42.71D6 in GAP, Magma, Sage, TeX
C_4^2._{71}D_6 % in TeX
G:=Group("C4^2.71D6"); // GroupNames label
G:=SmallGroup(192,628);
// by ID
G=gap.SmallGroup(192,628);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,253,120,254,555,100,1123,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^6=a^2,d^2=a^2*b,a*b=b*a,c*a*c^-1=a^-1*b^2,d*a*d^-1=a*b^2,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b^-1*c^5>;
// generators/relations
Export