metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24⋊4Q8, C8⋊3Dic6, C3⋊C8⋊2Q8, C3⋊3(C8⋊Q8), C4⋊C4.44D6, (C2×C8).64D6, C12⋊Q8.9C2, C4.27(S3×Q8), C2.D8.8S3, C6.17(C4⋊Q8), C12.59(C2×Q8), C24⋊C4.3C2, C2.12(C12⋊Q8), C8⋊Dic3.10C2, C4.24(C2×Dic6), C6.Q16.9C2, C2.21(D8⋊S3), C6.39(C8⋊C22), (C2×Dic3).46D4, C22.225(S3×D4), C4.Dic6.7C2, C12.Q8.7C2, (C2×C24).142C22, (C2×C12).292C23, C2.20(Q16⋊S3), C6.67(C8.C22), C4⋊Dic3.118C22, (C4×Dic3).35C22, (C3×C2.D8).7C2, (C2×C6).297(C2×D4), (C2×C3⋊C8).66C22, (C3×C4⋊C4).85C22, (C2×C4).395(C22×S3), SmallGroup(192,435)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24⋊4Q8
G = < a,b,c | a24=b4=1, c2=b2, bab-1=a7, cac-1=a5, cbc-1=b-1 >
Subgroups: 240 in 90 conjugacy classes, 43 normal (37 characteristic)
C1, C2, C3, C4, C4, C22, C6, C8, C8, C2×C4, C2×C4, Q8, Dic3, C12, C12, C2×C6, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×Q8, C3⋊C8, C24, Dic6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C8⋊C4, C4.Q8, C2.D8, C2.D8, C42.C2, C4⋊Q8, C2×C3⋊C8, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, C3×C4⋊C4, C2×C24, C2×Dic6, C8⋊Q8, C6.Q16, C12.Q8, C24⋊C4, C8⋊Dic3, C3×C2.D8, C12⋊Q8, C4.Dic6, C24⋊4Q8
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2×D4, C2×Q8, Dic6, C22×S3, C4⋊Q8, C8⋊C22, C8.C22, C2×Dic6, S3×D4, S3×Q8, C8⋊Q8, C12⋊Q8, D8⋊S3, Q16⋊S3, C24⋊4Q8
Character table of C24⋊4Q8
| 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 | 24A | 24B | 24C | 24D | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 8 | 8 | 12 | 12 | 24 | 24 | 2 | 2 | 2 | 4 | 4 | 12 | 12 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | |
| 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 | 0 | 0 | -2 | 2 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 2 | -2 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | 2 | 2 | -1 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ12 | 2 | 2 | 2 | 2 | -1 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 2 | 2 | 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 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -2 | -2 | 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 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -2 | -2 | 0 | 0 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ15 | 2 | -2 | -2 | 2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 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 | -2 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 2 | 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 | 0 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 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 | 2 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | -2 | symplectic lifted from Q8, Schur index 2 |
| ρ19 | 2 | -2 | -2 | 2 | -1 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | 2 | -2 | 0 | 0 | -1 | 1 | √3 | -√3 | -√3 | √3 | -1 | 1 | 1 | -1 | symplectic lifted from Dic6, Schur index 2 |
| ρ20 | 2 | -2 | -2 | 2 | -1 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | 2 | -2 | 0 | 0 | -1 | 1 | -√3 | √3 | √3 | -√3 | -1 | 1 | 1 | -1 | symplectic lifted from Dic6, Schur index 2 |
| ρ21 | 2 | -2 | -2 | 2 | -1 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -2 | 2 | 0 | 0 | -1 | 1 | -√3 | √3 | -√3 | √3 | 1 | -1 | -1 | 1 | symplectic lifted from Dic6, Schur index 2 |
| ρ22 | 2 | -2 | -2 | 2 | -1 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -2 | 2 | 0 | 0 | -1 | 1 | √3 | -√3 | √3 | -√3 | 1 | -1 | -1 | 1 | symplectic lifted from Dic6, Schur index 2 |
| ρ23 | 4 | 4 | 4 | 4 | -2 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3×D4 |
| 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 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from S3×Q8, 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 | 0 | 0 | 0 | 0 | 0 | 0 | √-6 | -√-6 | √-6 | -√-6 | complex lifted from Q16⋊S3 |
| ρ28 | 4 | 4 | -4 | -4 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√-6 | √-6 | -√-6 | √-6 | complex lifted from Q16⋊S3 |
| 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 | 0 | 0 | √-6 | √-6 | -√-6 | -√-6 | complex lifted from D8⋊S3 |
| 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 | 0 | 0 | -√-6 | -√-6 | √-6 | √-6 | complex lifted from D8⋊S3 |
►Regular action on 192 points
Generators in S192
(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 63 115 27)(2 70 116 34)(3 53 117 41)(4 60 118 48)(5 67 119 31)(6 50 120 38)(7 57 97 45)(8 64 98 28)(9 71 99 35)(10 54 100 42)(11 61 101 25)(12 68 102 32)(13 51 103 39)(14 58 104 46)(15 65 105 29)(16 72 106 36)(17 55 107 43)(18 62 108 26)(19 69 109 33)(20 52 110 40)(21 59 111 47)(22 66 112 30)(23 49 113 37)(24 56 114 44)(73 135 159 188)(74 142 160 171)(75 125 161 178)(76 132 162 185)(77 139 163 192)(78 122 164 175)(79 129 165 182)(80 136 166 189)(81 143 167 172)(82 126 168 179)(83 133 145 186)(84 140 146 169)(85 123 147 176)(86 130 148 183)(87 137 149 190)(88 144 150 173)(89 127 151 180)(90 134 152 187)(91 141 153 170)(92 124 154 177)(93 131 155 184)(94 138 156 191)(95 121 157 174)(96 128 158 181)
(1 176 115 123)(2 181 116 128)(3 186 117 133)(4 191 118 138)(5 172 119 143)(6 177 120 124)(7 182 97 129)(8 187 98 134)(9 192 99 139)(10 173 100 144)(11 178 101 125)(12 183 102 130)(13 188 103 135)(14 169 104 140)(15 174 105 121)(16 179 106 126)(17 184 107 131)(18 189 108 136)(19 170 109 141)(20 175 110 122)(21 180 111 127)(22 185 112 132)(23 190 113 137)(24 171 114 142)(25 75 61 161)(26 80 62 166)(27 85 63 147)(28 90 64 152)(29 95 65 157)(30 76 66 162)(31 81 67 167)(32 86 68 148)(33 91 69 153)(34 96 70 158)(35 77 71 163)(36 82 72 168)(37 87 49 149)(38 92 50 154)(39 73 51 159)(40 78 52 164)(41 83 53 145)(42 88 54 150)(43 93 55 155)(44 74 56 160)(45 79 57 165)(46 84 58 146)(47 89 59 151)(48 94 60 156)
G:=sub<Sym(192)| (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), (1,63,115,27)(2,70,116,34)(3,53,117,41)(4,60,118,48)(5,67,119,31)(6,50,120,38)(7,57,97,45)(8,64,98,28)(9,71,99,35)(10,54,100,42)(11,61,101,25)(12,68,102,32)(13,51,103,39)(14,58,104,46)(15,65,105,29)(16,72,106,36)(17,55,107,43)(18,62,108,26)(19,69,109,33)(20,52,110,40)(21,59,111,47)(22,66,112,30)(23,49,113,37)(24,56,114,44)(73,135,159,188)(74,142,160,171)(75,125,161,178)(76,132,162,185)(77,139,163,192)(78,122,164,175)(79,129,165,182)(80,136,166,189)(81,143,167,172)(82,126,168,179)(83,133,145,186)(84,140,146,169)(85,123,147,176)(86,130,148,183)(87,137,149,190)(88,144,150,173)(89,127,151,180)(90,134,152,187)(91,141,153,170)(92,124,154,177)(93,131,155,184)(94,138,156,191)(95,121,157,174)(96,128,158,181), (1,176,115,123)(2,181,116,128)(3,186,117,133)(4,191,118,138)(5,172,119,143)(6,177,120,124)(7,182,97,129)(8,187,98,134)(9,192,99,139)(10,173,100,144)(11,178,101,125)(12,183,102,130)(13,188,103,135)(14,169,104,140)(15,174,105,121)(16,179,106,126)(17,184,107,131)(18,189,108,136)(19,170,109,141)(20,175,110,122)(21,180,111,127)(22,185,112,132)(23,190,113,137)(24,171,114,142)(25,75,61,161)(26,80,62,166)(27,85,63,147)(28,90,64,152)(29,95,65,157)(30,76,66,162)(31,81,67,167)(32,86,68,148)(33,91,69,153)(34,96,70,158)(35,77,71,163)(36,82,72,168)(37,87,49,149)(38,92,50,154)(39,73,51,159)(40,78,52,164)(41,83,53,145)(42,88,54,150)(43,93,55,155)(44,74,56,160)(45,79,57,165)(46,84,58,146)(47,89,59,151)(48,94,60,156)>;
G:=Group( (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), (1,63,115,27)(2,70,116,34)(3,53,117,41)(4,60,118,48)(5,67,119,31)(6,50,120,38)(7,57,97,45)(8,64,98,28)(9,71,99,35)(10,54,100,42)(11,61,101,25)(12,68,102,32)(13,51,103,39)(14,58,104,46)(15,65,105,29)(16,72,106,36)(17,55,107,43)(18,62,108,26)(19,69,109,33)(20,52,110,40)(21,59,111,47)(22,66,112,30)(23,49,113,37)(24,56,114,44)(73,135,159,188)(74,142,160,171)(75,125,161,178)(76,132,162,185)(77,139,163,192)(78,122,164,175)(79,129,165,182)(80,136,166,189)(81,143,167,172)(82,126,168,179)(83,133,145,186)(84,140,146,169)(85,123,147,176)(86,130,148,183)(87,137,149,190)(88,144,150,173)(89,127,151,180)(90,134,152,187)(91,141,153,170)(92,124,154,177)(93,131,155,184)(94,138,156,191)(95,121,157,174)(96,128,158,181), (1,176,115,123)(2,181,116,128)(3,186,117,133)(4,191,118,138)(5,172,119,143)(6,177,120,124)(7,182,97,129)(8,187,98,134)(9,192,99,139)(10,173,100,144)(11,178,101,125)(12,183,102,130)(13,188,103,135)(14,169,104,140)(15,174,105,121)(16,179,106,126)(17,184,107,131)(18,189,108,136)(19,170,109,141)(20,175,110,122)(21,180,111,127)(22,185,112,132)(23,190,113,137)(24,171,114,142)(25,75,61,161)(26,80,62,166)(27,85,63,147)(28,90,64,152)(29,95,65,157)(30,76,66,162)(31,81,67,167)(32,86,68,148)(33,91,69,153)(34,96,70,158)(35,77,71,163)(36,82,72,168)(37,87,49,149)(38,92,50,154)(39,73,51,159)(40,78,52,164)(41,83,53,145)(42,88,54,150)(43,93,55,155)(44,74,56,160)(45,79,57,165)(46,84,58,146)(47,89,59,151)(48,94,60,156) );
G=PermutationGroup([[(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)], [(1,63,115,27),(2,70,116,34),(3,53,117,41),(4,60,118,48),(5,67,119,31),(6,50,120,38),(7,57,97,45),(8,64,98,28),(9,71,99,35),(10,54,100,42),(11,61,101,25),(12,68,102,32),(13,51,103,39),(14,58,104,46),(15,65,105,29),(16,72,106,36),(17,55,107,43),(18,62,108,26),(19,69,109,33),(20,52,110,40),(21,59,111,47),(22,66,112,30),(23,49,113,37),(24,56,114,44),(73,135,159,188),(74,142,160,171),(75,125,161,178),(76,132,162,185),(77,139,163,192),(78,122,164,175),(79,129,165,182),(80,136,166,189),(81,143,167,172),(82,126,168,179),(83,133,145,186),(84,140,146,169),(85,123,147,176),(86,130,148,183),(87,137,149,190),(88,144,150,173),(89,127,151,180),(90,134,152,187),(91,141,153,170),(92,124,154,177),(93,131,155,184),(94,138,156,191),(95,121,157,174),(96,128,158,181)], [(1,176,115,123),(2,181,116,128),(3,186,117,133),(4,191,118,138),(5,172,119,143),(6,177,120,124),(7,182,97,129),(8,187,98,134),(9,192,99,139),(10,173,100,144),(11,178,101,125),(12,183,102,130),(13,188,103,135),(14,169,104,140),(15,174,105,121),(16,179,106,126),(17,184,107,131),(18,189,108,136),(19,170,109,141),(20,175,110,122),(21,180,111,127),(22,185,112,132),(23,190,113,137),(24,171,114,142),(25,75,61,161),(26,80,62,166),(27,85,63,147),(28,90,64,152),(29,95,65,157),(30,76,66,162),(31,81,67,167),(32,86,68,148),(33,91,69,153),(34,96,70,158),(35,77,71,163),(36,82,72,168),(37,87,49,149),(38,92,50,154),(39,73,51,159),(40,78,52,164),(41,83,53,145),(42,88,54,150),(43,93,55,155),(44,74,56,160),(45,79,57,165),(46,84,58,146),(47,89,59,151),(48,94,60,156)]])
Matrix representation of C24⋊4Q8 ►in GL6(𝔽73)
| 61 | 72 | 0 | 0 | 0 | 0 |
| 72 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 42 | 31 | 42 | 31 |
| 0 | 0 | 42 | 11 | 42 | 11 |
| 0 | 0 | 31 | 42 | 42 | 31 |
| 0 | 0 | 31 | 62 | 42 | 11 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 12 | 21 | 28 |
| 0 | 0 | 61 | 2 | 45 | 66 |
| 0 | 0 | 21 | 28 | 59 | 61 |
| 0 | 0 | 45 | 66 | 12 | 71 |
| 61 | 72 | 0 | 0 | 0 | 0 |
| 72 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 25 | 41 | 28 | 34 |
| 0 | 0 | 16 | 48 | 6 | 45 |
| 0 | 0 | 45 | 39 | 25 | 41 |
| 0 | 0 | 67 | 28 | 16 | 48 |
G:=sub<GL(6,GF(73))| [61,72,0,0,0,0,72,12,0,0,0,0,0,0,42,42,31,31,0,0,31,11,42,62,0,0,42,42,42,42,0,0,31,11,31,11],[0,1,0,0,0,0,72,0,0,0,0,0,0,0,14,61,21,45,0,0,12,2,28,66,0,0,21,45,59,12,0,0,28,66,61,71],[61,72,0,0,0,0,72,12,0,0,0,0,0,0,25,16,45,67,0,0,41,48,39,28,0,0,28,6,25,16,0,0,34,45,41,48] >;
C24⋊4Q8 in GAP, Magma, Sage, TeX
C_{24}\rtimes_4Q_8 % in TeX
G:=Group("C24:4Q8"); // GroupNames label
G:=SmallGroup(192,435);
// by ID
G=gap.SmallGroup(192,435);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,477,120,254,219,58,438,102,6278]);
// Polycyclic
G:=Group<a,b,c|a^24=b^4=1,c^2=b^2,b*a*b^-1=a^7,c*a*c^-1=a^5,c*b*c^-1=b^-1>;
// generators/relations
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