metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.77D6, C4⋊Q8.8S3, (C2×Q8).67D6, (C2×C12).294D4, C12.79(C4○D4), C12.6Q8.9C2, (C6×Q8).61C22, C4.25(D4⋊2S3), (C4×C12).129C22, (C2×C12).400C23, Q8⋊2Dic3.12C2, C6.46(C4.4D4), C6.94(C8.C22), C42.S3.8C2, C4⋊Dic3.160C22, C2.13(C23.12D6), C2.15(Q8.11D6), C3⋊4(C42.30C22), (C3×C4⋊Q8).8C2, (C2×C6).531(C2×D4), (C2×C4).72(C3⋊D4), (C2×C3⋊C8).134C22, (C2×C4).497(C22×S3), C22.203(C2×C3⋊D4), SmallGroup(192,641)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.77D6
G = < a,b,c,d | a4=b4=1, c6=b2, d2=a2b2, ab=ba, cac-1=a-1, dad-1=a-1b2, cbc-1=dbd-1=b-1, dcd-1=a2bc5 >
Subgroups: 208 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, C2×C8, C2×Q8, C3⋊C8, C2×Dic3, C2×C12, C2×C12, C2×C12, C3×Q8, C8⋊C4, Q8⋊C4, C42.C2, C4⋊Q8, C2×C3⋊C8, Dic3⋊C4, C4⋊Dic3, C4×C12, C3×C4⋊C4, C6×Q8, C42.30C22, C42.S3, Q8⋊2Dic3, C12.6Q8, C3×C4⋊Q8, C42.77D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C3⋊D4, C22×S3, C4.4D4, C8.C22, D4⋊2S3, C2×C3⋊D4, C42.30C22, C23.12D6, Q8.11D6, C42.77D6
Character table of C42.77D6
| 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 | 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 |
| 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 | 0 | -2 | 0 | 0 | 2 | 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 | 0 | -2 | 0 | 0 | 2 | 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 | 0 | 2 | 0 | 0 | -2 | 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 | 0 | 2 | 0 | 0 | -2 | 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 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | symplectic lifted from D4⋊2S3, Schur index 2 |
| 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 | symplectic lifted from C8.C22, 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 D4⋊2S3, 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 Q8.11D6 |
| 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 Q8.11D6 |
| 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 92 46 151)(2 152 47 93)(3 94 48 153)(4 154 37 95)(5 96 38 155)(6 156 39 85)(7 86 40 145)(8 146 41 87)(9 88 42 147)(10 148 43 89)(11 90 44 149)(12 150 45 91)(13 53 64 105)(14 106 65 54)(15 55 66 107)(16 108 67 56)(17 57 68 97)(18 98 69 58)(19 59 70 99)(20 100 71 60)(21 49 72 101)(22 102 61 50)(23 51 62 103)(24 104 63 52)(25 165 141 116)(26 117 142 166)(27 167 143 118)(28 119 144 168)(29 157 133 120)(30 109 134 158)(31 159 135 110)(32 111 136 160)(33 161 137 112)(34 113 138 162)(35 163 139 114)(36 115 140 164)(73 192 180 123)(74 124 169 181)(75 182 170 125)(76 126 171 183)(77 184 172 127)(78 128 173 185)(79 186 174 129)(80 130 175 187)(81 188 176 131)(82 132 177 189)(83 190 178 121)(84 122 179 191)
(1 143 7 137)(2 138 8 144)(3 133 9 139)(4 140 10 134)(5 135 11 141)(6 142 12 136)(13 131 19 125)(14 126 20 132)(15 121 21 127)(16 128 22 122)(17 123 23 129)(18 130 24 124)(25 38 31 44)(26 45 32 39)(27 40 33 46)(28 47 34 41)(29 42 35 48)(30 37 36 43)(49 77 55 83)(50 84 56 78)(51 79 57 73)(52 74 58 80)(53 81 59 75)(54 76 60 82)(61 191 67 185)(62 186 68 192)(63 181 69 187)(64 188 70 182)(65 183 71 189)(66 190 72 184)(85 117 91 111)(86 112 92 118)(87 119 93 113)(88 114 94 120)(89 109 95 115)(90 116 96 110)(97 180 103 174)(98 175 104 169)(99 170 105 176)(100 177 106 171)(101 172 107 178)(102 179 108 173)(145 161 151 167)(146 168 152 162)(147 163 153 157)(148 158 154 164)(149 165 155 159)(150 160 156 166)
(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 80 40 169)(2 97 41 51)(3 78 42 179)(4 107 43 49)(5 76 44 177)(6 105 45 59)(7 74 46 175)(8 103 47 57)(9 84 48 173)(10 101 37 55)(11 82 38 171)(12 99 39 53)(13 156 70 91)(14 110 71 165)(15 154 72 89)(16 120 61 163)(17 152 62 87)(18 118 63 161)(19 150 64 85)(20 116 65 159)(21 148 66 95)(22 114 67 157)(23 146 68 93)(24 112 69 167)(25 100 135 54)(26 75 136 176)(27 98 137 52)(28 73 138 174)(29 108 139 50)(30 83 140 172)(31 106 141 60)(32 81 142 170)(33 104 143 58)(34 79 144 180)(35 102 133 56)(36 77 134 178)(86 187 151 124)(88 185 153 122)(90 183 155 132)(92 181 145 130)(94 191 147 128)(96 189 149 126)(109 127 164 190)(111 125 166 188)(113 123 168 186)(115 121 158 184)(117 131 160 182)(119 129 162 192)
G:=sub<Sym(192)| (1,92,46,151)(2,152,47,93)(3,94,48,153)(4,154,37,95)(5,96,38,155)(6,156,39,85)(7,86,40,145)(8,146,41,87)(9,88,42,147)(10,148,43,89)(11,90,44,149)(12,150,45,91)(13,53,64,105)(14,106,65,54)(15,55,66,107)(16,108,67,56)(17,57,68,97)(18,98,69,58)(19,59,70,99)(20,100,71,60)(21,49,72,101)(22,102,61,50)(23,51,62,103)(24,104,63,52)(25,165,141,116)(26,117,142,166)(27,167,143,118)(28,119,144,168)(29,157,133,120)(30,109,134,158)(31,159,135,110)(32,111,136,160)(33,161,137,112)(34,113,138,162)(35,163,139,114)(36,115,140,164)(73,192,180,123)(74,124,169,181)(75,182,170,125)(76,126,171,183)(77,184,172,127)(78,128,173,185)(79,186,174,129)(80,130,175,187)(81,188,176,131)(82,132,177,189)(83,190,178,121)(84,122,179,191), (1,143,7,137)(2,138,8,144)(3,133,9,139)(4,140,10,134)(5,135,11,141)(6,142,12,136)(13,131,19,125)(14,126,20,132)(15,121,21,127)(16,128,22,122)(17,123,23,129)(18,130,24,124)(25,38,31,44)(26,45,32,39)(27,40,33,46)(28,47,34,41)(29,42,35,48)(30,37,36,43)(49,77,55,83)(50,84,56,78)(51,79,57,73)(52,74,58,80)(53,81,59,75)(54,76,60,82)(61,191,67,185)(62,186,68,192)(63,181,69,187)(64,188,70,182)(65,183,71,189)(66,190,72,184)(85,117,91,111)(86,112,92,118)(87,119,93,113)(88,114,94,120)(89,109,95,115)(90,116,96,110)(97,180,103,174)(98,175,104,169)(99,170,105,176)(100,177,106,171)(101,172,107,178)(102,179,108,173)(145,161,151,167)(146,168,152,162)(147,163,153,157)(148,158,154,164)(149,165,155,159)(150,160,156,166), (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), (1,80,40,169)(2,97,41,51)(3,78,42,179)(4,107,43,49)(5,76,44,177)(6,105,45,59)(7,74,46,175)(8,103,47,57)(9,84,48,173)(10,101,37,55)(11,82,38,171)(12,99,39,53)(13,156,70,91)(14,110,71,165)(15,154,72,89)(16,120,61,163)(17,152,62,87)(18,118,63,161)(19,150,64,85)(20,116,65,159)(21,148,66,95)(22,114,67,157)(23,146,68,93)(24,112,69,167)(25,100,135,54)(26,75,136,176)(27,98,137,52)(28,73,138,174)(29,108,139,50)(30,83,140,172)(31,106,141,60)(32,81,142,170)(33,104,143,58)(34,79,144,180)(35,102,133,56)(36,77,134,178)(86,187,151,124)(88,185,153,122)(90,183,155,132)(92,181,145,130)(94,191,147,128)(96,189,149,126)(109,127,164,190)(111,125,166,188)(113,123,168,186)(115,121,158,184)(117,131,160,182)(119,129,162,192)>;
G:=Group( (1,92,46,151)(2,152,47,93)(3,94,48,153)(4,154,37,95)(5,96,38,155)(6,156,39,85)(7,86,40,145)(8,146,41,87)(9,88,42,147)(10,148,43,89)(11,90,44,149)(12,150,45,91)(13,53,64,105)(14,106,65,54)(15,55,66,107)(16,108,67,56)(17,57,68,97)(18,98,69,58)(19,59,70,99)(20,100,71,60)(21,49,72,101)(22,102,61,50)(23,51,62,103)(24,104,63,52)(25,165,141,116)(26,117,142,166)(27,167,143,118)(28,119,144,168)(29,157,133,120)(30,109,134,158)(31,159,135,110)(32,111,136,160)(33,161,137,112)(34,113,138,162)(35,163,139,114)(36,115,140,164)(73,192,180,123)(74,124,169,181)(75,182,170,125)(76,126,171,183)(77,184,172,127)(78,128,173,185)(79,186,174,129)(80,130,175,187)(81,188,176,131)(82,132,177,189)(83,190,178,121)(84,122,179,191), (1,143,7,137)(2,138,8,144)(3,133,9,139)(4,140,10,134)(5,135,11,141)(6,142,12,136)(13,131,19,125)(14,126,20,132)(15,121,21,127)(16,128,22,122)(17,123,23,129)(18,130,24,124)(25,38,31,44)(26,45,32,39)(27,40,33,46)(28,47,34,41)(29,42,35,48)(30,37,36,43)(49,77,55,83)(50,84,56,78)(51,79,57,73)(52,74,58,80)(53,81,59,75)(54,76,60,82)(61,191,67,185)(62,186,68,192)(63,181,69,187)(64,188,70,182)(65,183,71,189)(66,190,72,184)(85,117,91,111)(86,112,92,118)(87,119,93,113)(88,114,94,120)(89,109,95,115)(90,116,96,110)(97,180,103,174)(98,175,104,169)(99,170,105,176)(100,177,106,171)(101,172,107,178)(102,179,108,173)(145,161,151,167)(146,168,152,162)(147,163,153,157)(148,158,154,164)(149,165,155,159)(150,160,156,166), (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), (1,80,40,169)(2,97,41,51)(3,78,42,179)(4,107,43,49)(5,76,44,177)(6,105,45,59)(7,74,46,175)(8,103,47,57)(9,84,48,173)(10,101,37,55)(11,82,38,171)(12,99,39,53)(13,156,70,91)(14,110,71,165)(15,154,72,89)(16,120,61,163)(17,152,62,87)(18,118,63,161)(19,150,64,85)(20,116,65,159)(21,148,66,95)(22,114,67,157)(23,146,68,93)(24,112,69,167)(25,100,135,54)(26,75,136,176)(27,98,137,52)(28,73,138,174)(29,108,139,50)(30,83,140,172)(31,106,141,60)(32,81,142,170)(33,104,143,58)(34,79,144,180)(35,102,133,56)(36,77,134,178)(86,187,151,124)(88,185,153,122)(90,183,155,132)(92,181,145,130)(94,191,147,128)(96,189,149,126)(109,127,164,190)(111,125,166,188)(113,123,168,186)(115,121,158,184)(117,131,160,182)(119,129,162,192) );
G=PermutationGroup([[(1,92,46,151),(2,152,47,93),(3,94,48,153),(4,154,37,95),(5,96,38,155),(6,156,39,85),(7,86,40,145),(8,146,41,87),(9,88,42,147),(10,148,43,89),(11,90,44,149),(12,150,45,91),(13,53,64,105),(14,106,65,54),(15,55,66,107),(16,108,67,56),(17,57,68,97),(18,98,69,58),(19,59,70,99),(20,100,71,60),(21,49,72,101),(22,102,61,50),(23,51,62,103),(24,104,63,52),(25,165,141,116),(26,117,142,166),(27,167,143,118),(28,119,144,168),(29,157,133,120),(30,109,134,158),(31,159,135,110),(32,111,136,160),(33,161,137,112),(34,113,138,162),(35,163,139,114),(36,115,140,164),(73,192,180,123),(74,124,169,181),(75,182,170,125),(76,126,171,183),(77,184,172,127),(78,128,173,185),(79,186,174,129),(80,130,175,187),(81,188,176,131),(82,132,177,189),(83,190,178,121),(84,122,179,191)], [(1,143,7,137),(2,138,8,144),(3,133,9,139),(4,140,10,134),(5,135,11,141),(6,142,12,136),(13,131,19,125),(14,126,20,132),(15,121,21,127),(16,128,22,122),(17,123,23,129),(18,130,24,124),(25,38,31,44),(26,45,32,39),(27,40,33,46),(28,47,34,41),(29,42,35,48),(30,37,36,43),(49,77,55,83),(50,84,56,78),(51,79,57,73),(52,74,58,80),(53,81,59,75),(54,76,60,82),(61,191,67,185),(62,186,68,192),(63,181,69,187),(64,188,70,182),(65,183,71,189),(66,190,72,184),(85,117,91,111),(86,112,92,118),(87,119,93,113),(88,114,94,120),(89,109,95,115),(90,116,96,110),(97,180,103,174),(98,175,104,169),(99,170,105,176),(100,177,106,171),(101,172,107,178),(102,179,108,173),(145,161,151,167),(146,168,152,162),(147,163,153,157),(148,158,154,164),(149,165,155,159),(150,160,156,166)], [(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)], [(1,80,40,169),(2,97,41,51),(3,78,42,179),(4,107,43,49),(5,76,44,177),(6,105,45,59),(7,74,46,175),(8,103,47,57),(9,84,48,173),(10,101,37,55),(11,82,38,171),(12,99,39,53),(13,156,70,91),(14,110,71,165),(15,154,72,89),(16,120,61,163),(17,152,62,87),(18,118,63,161),(19,150,64,85),(20,116,65,159),(21,148,66,95),(22,114,67,157),(23,146,68,93),(24,112,69,167),(25,100,135,54),(26,75,136,176),(27,98,137,52),(28,73,138,174),(29,108,139,50),(30,83,140,172),(31,106,141,60),(32,81,142,170),(33,104,143,58),(34,79,144,180),(35,102,133,56),(36,77,134,178),(86,187,151,124),(88,185,153,122),(90,183,155,132),(92,181,145,130),(94,191,147,128),(96,189,149,126),(109,127,164,190),(111,125,166,188),(113,123,168,186),(115,121,158,184),(117,131,160,182),(119,129,162,192)]])
Matrix representation of C42.77D6 ►in GL8(𝔽73)
| 22 | 8 | 35 | 1 | 0 | 0 | 0 | 0 |
| 65 | 14 | 72 | 34 | 0 | 0 | 0 | 0 |
| 38 | 72 | 51 | 65 | 0 | 0 | 0 | 0 |
| 1 | 39 | 8 | 59 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 30 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 43 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 30 | 60 |
| 0 | 0 | 0 | 0 | 0 | 0 | 13 | 43 |
| 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 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 72 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 72 | 72 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 68 | 9 | 26 |
| 0 | 0 | 0 | 0 | 5 | 11 | 47 | 56 |
| 0 | 0 | 0 | 0 | 9 | 26 | 67 | 5 |
| 0 | 0 | 0 | 0 | 47 | 56 | 68 | 62 |
| 36 | 6 | 41 | 19 | 0 | 0 | 0 | 0 |
| 43 | 37 | 51 | 32 | 0 | 0 | 0 | 0 |
| 32 | 54 | 37 | 67 | 0 | 0 | 0 | 0 |
| 22 | 41 | 30 | 36 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 38 | 2 | 7 | 58 |
| 0 | 0 | 0 | 0 | 37 | 35 | 51 | 66 |
| 0 | 0 | 0 | 0 | 7 | 58 | 35 | 71 |
| 0 | 0 | 0 | 0 | 51 | 66 | 36 | 38 |
G:=sub<GL(8,GF(73))| [22,65,38,1,0,0,0,0,8,14,72,39,0,0,0,0,35,72,51,8,0,0,0,0,1,34,65,59,0,0,0,0,0,0,0,0,30,13,0,0,0,0,0,0,60,43,0,0,0,0,0,0,0,0,30,13,0,0,0,0,0,0,60,43],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[0,0,72,1,0,0,0,0,0,0,72,0,0,0,0,0,72,1,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,0,6,5,9,47,0,0,0,0,68,11,26,56,0,0,0,0,9,47,67,68,0,0,0,0,26,56,5,62],[36,43,32,22,0,0,0,0,6,37,54,41,0,0,0,0,41,51,37,30,0,0,0,0,19,32,67,36,0,0,0,0,0,0,0,0,38,37,7,51,0,0,0,0,2,35,58,66,0,0,0,0,7,51,35,36,0,0,0,0,58,66,71,38] >;
C42.77D6 in GAP, Magma, Sage, TeX
C_4^2._{77}D_6 % in TeX
G:=Group("C4^2.77D6"); // GroupNames label
G:=SmallGroup(192,641);
// by ID
G=gap.SmallGroup(192,641);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,477,64,590,135,184,438,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^6=b^2,d^2=a^2*b^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=a^2*b*c^5>;
// generators/relations
Export