metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24⋊8Q8, C8⋊7Dic6, C4.4D24, C12.29D8, C12.15Q16, C4.4Dic12, C42.254D6, (C4×C8).7S3, C6.1(C2×D8), (C4×C24).9C2, C2.4(C2×D24), C3⋊1(C8⋊2Q8), C6.3(C4⋊Q8), C6.2(C2×Q16), (C2×C4).78D12, (C2×C8).300D6, C12.70(C2×Q8), C24⋊1C4.4C2, (C2×C12).375D4, C12⋊2Q8.3C2, C4.36(C2×Dic6), C2.5(C2×Dic12), C2.7(C12⋊2Q8), C22.87(C2×D12), C4⋊Dic3.3C22, (C2×C12).718C23, (C4×C12).304C22, (C2×C24).373C22, (C2×C6).101(C2×D4), (C2×C4).661(C22×S3), SmallGroup(192,241)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24⋊8Q8
G = < a,b,c | a24=b4=1, c2=b2, ab=ba, cac-1=a-1, cbc-1=b-1 >
Subgroups: 280 in 98 conjugacy classes, 55 normal (21 characteristic)
C1, C2, C3, C4, C4, C22, C6, C8, C2×C4, C2×C4, Q8, Dic3, C12, C2×C6, C42, C4⋊C4, C2×C8, C2×Q8, C24, Dic6, C2×Dic3, C2×C12, C4×C8, C2.D8, C4⋊Q8, C4⋊Dic3, C4⋊Dic3, C4×C12, C2×C24, C2×Dic6, C8⋊2Q8, C24⋊1C4, C4×C24, C12⋊2Q8, C24⋊8Q8
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, D8, Q16, C2×D4, C2×Q8, Dic6, D12, C22×S3, C4⋊Q8, C2×D8, C2×Q16, D24, Dic12, C2×Dic6, C2×D12, C8⋊2Q8, C12⋊2Q8, C2×D24, C2×Dic12, C24⋊8Q8
►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)
(1 142 63 176)(2 143 64 177)(3 144 65 178)(4 121 66 179)(5 122 67 180)(6 123 68 181)(7 124 69 182)(8 125 70 183)(9 126 71 184)(10 127 72 185)(11 128 49 186)(12 129 50 187)(13 130 51 188)(14 131 52 189)(15 132 53 190)(16 133 54 191)(17 134 55 192)(18 135 56 169)(19 136 57 170)(20 137 58 171)(21 138 59 172)(22 139 60 173)(23 140 61 174)(24 141 62 175)(25 152 86 108)(26 153 87 109)(27 154 88 110)(28 155 89 111)(29 156 90 112)(30 157 91 113)(31 158 92 114)(32 159 93 115)(33 160 94 116)(34 161 95 117)(35 162 96 118)(36 163 73 119)(37 164 74 120)(38 165 75 97)(39 166 76 98)(40 167 77 99)(41 168 78 100)(42 145 79 101)(43 146 80 102)(44 147 81 103)(45 148 82 104)(46 149 83 105)(47 150 84 106)(48 151 85 107)
(1 103 63 147)(2 102 64 146)(3 101 65 145)(4 100 66 168)(5 99 67 167)(6 98 68 166)(7 97 69 165)(8 120 70 164)(9 119 71 163)(10 118 72 162)(11 117 49 161)(12 116 50 160)(13 115 51 159)(14 114 52 158)(15 113 53 157)(16 112 54 156)(17 111 55 155)(18 110 56 154)(19 109 57 153)(20 108 58 152)(21 107 59 151)(22 106 60 150)(23 105 61 149)(24 104 62 148)(25 137 86 171)(26 136 87 170)(27 135 88 169)(28 134 89 192)(29 133 90 191)(30 132 91 190)(31 131 92 189)(32 130 93 188)(33 129 94 187)(34 128 95 186)(35 127 96 185)(36 126 73 184)(37 125 74 183)(38 124 75 182)(39 123 76 181)(40 122 77 180)(41 121 78 179)(42 144 79 178)(43 143 80 177)(44 142 81 176)(45 141 82 175)(46 140 83 174)(47 139 84 173)(48 138 85 172)
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), (1,142,63,176)(2,143,64,177)(3,144,65,178)(4,121,66,179)(5,122,67,180)(6,123,68,181)(7,124,69,182)(8,125,70,183)(9,126,71,184)(10,127,72,185)(11,128,49,186)(12,129,50,187)(13,130,51,188)(14,131,52,189)(15,132,53,190)(16,133,54,191)(17,134,55,192)(18,135,56,169)(19,136,57,170)(20,137,58,171)(21,138,59,172)(22,139,60,173)(23,140,61,174)(24,141,62,175)(25,152,86,108)(26,153,87,109)(27,154,88,110)(28,155,89,111)(29,156,90,112)(30,157,91,113)(31,158,92,114)(32,159,93,115)(33,160,94,116)(34,161,95,117)(35,162,96,118)(36,163,73,119)(37,164,74,120)(38,165,75,97)(39,166,76,98)(40,167,77,99)(41,168,78,100)(42,145,79,101)(43,146,80,102)(44,147,81,103)(45,148,82,104)(46,149,83,105)(47,150,84,106)(48,151,85,107), (1,103,63,147)(2,102,64,146)(3,101,65,145)(4,100,66,168)(5,99,67,167)(6,98,68,166)(7,97,69,165)(8,120,70,164)(9,119,71,163)(10,118,72,162)(11,117,49,161)(12,116,50,160)(13,115,51,159)(14,114,52,158)(15,113,53,157)(16,112,54,156)(17,111,55,155)(18,110,56,154)(19,109,57,153)(20,108,58,152)(21,107,59,151)(22,106,60,150)(23,105,61,149)(24,104,62,148)(25,137,86,171)(26,136,87,170)(27,135,88,169)(28,134,89,192)(29,133,90,191)(30,132,91,190)(31,131,92,189)(32,130,93,188)(33,129,94,187)(34,128,95,186)(35,127,96,185)(36,126,73,184)(37,125,74,183)(38,124,75,182)(39,123,76,181)(40,122,77,180)(41,121,78,179)(42,144,79,178)(43,143,80,177)(44,142,81,176)(45,141,82,175)(46,140,83,174)(47,139,84,173)(48,138,85,172)>;
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), (1,142,63,176)(2,143,64,177)(3,144,65,178)(4,121,66,179)(5,122,67,180)(6,123,68,181)(7,124,69,182)(8,125,70,183)(9,126,71,184)(10,127,72,185)(11,128,49,186)(12,129,50,187)(13,130,51,188)(14,131,52,189)(15,132,53,190)(16,133,54,191)(17,134,55,192)(18,135,56,169)(19,136,57,170)(20,137,58,171)(21,138,59,172)(22,139,60,173)(23,140,61,174)(24,141,62,175)(25,152,86,108)(26,153,87,109)(27,154,88,110)(28,155,89,111)(29,156,90,112)(30,157,91,113)(31,158,92,114)(32,159,93,115)(33,160,94,116)(34,161,95,117)(35,162,96,118)(36,163,73,119)(37,164,74,120)(38,165,75,97)(39,166,76,98)(40,167,77,99)(41,168,78,100)(42,145,79,101)(43,146,80,102)(44,147,81,103)(45,148,82,104)(46,149,83,105)(47,150,84,106)(48,151,85,107), (1,103,63,147)(2,102,64,146)(3,101,65,145)(4,100,66,168)(5,99,67,167)(6,98,68,166)(7,97,69,165)(8,120,70,164)(9,119,71,163)(10,118,72,162)(11,117,49,161)(12,116,50,160)(13,115,51,159)(14,114,52,158)(15,113,53,157)(16,112,54,156)(17,111,55,155)(18,110,56,154)(19,109,57,153)(20,108,58,152)(21,107,59,151)(22,106,60,150)(23,105,61,149)(24,104,62,148)(25,137,86,171)(26,136,87,170)(27,135,88,169)(28,134,89,192)(29,133,90,191)(30,132,91,190)(31,131,92,189)(32,130,93,188)(33,129,94,187)(34,128,95,186)(35,127,96,185)(36,126,73,184)(37,125,74,183)(38,124,75,182)(39,123,76,181)(40,122,77,180)(41,121,78,179)(42,144,79,178)(43,143,80,177)(44,142,81,176)(45,141,82,175)(46,140,83,174)(47,139,84,173)(48,138,85,172) );
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)], [(1,142,63,176),(2,143,64,177),(3,144,65,178),(4,121,66,179),(5,122,67,180),(6,123,68,181),(7,124,69,182),(8,125,70,183),(9,126,71,184),(10,127,72,185),(11,128,49,186),(12,129,50,187),(13,130,51,188),(14,131,52,189),(15,132,53,190),(16,133,54,191),(17,134,55,192),(18,135,56,169),(19,136,57,170),(20,137,58,171),(21,138,59,172),(22,139,60,173),(23,140,61,174),(24,141,62,175),(25,152,86,108),(26,153,87,109),(27,154,88,110),(28,155,89,111),(29,156,90,112),(30,157,91,113),(31,158,92,114),(32,159,93,115),(33,160,94,116),(34,161,95,117),(35,162,96,118),(36,163,73,119),(37,164,74,120),(38,165,75,97),(39,166,76,98),(40,167,77,99),(41,168,78,100),(42,145,79,101),(43,146,80,102),(44,147,81,103),(45,148,82,104),(46,149,83,105),(47,150,84,106),(48,151,85,107)], [(1,103,63,147),(2,102,64,146),(3,101,65,145),(4,100,66,168),(5,99,67,167),(6,98,68,166),(7,97,69,165),(8,120,70,164),(9,119,71,163),(10,118,72,162),(11,117,49,161),(12,116,50,160),(13,115,51,159),(14,114,52,158),(15,113,53,157),(16,112,54,156),(17,111,55,155),(18,110,56,154),(19,109,57,153),(20,108,58,152),(21,107,59,151),(22,106,60,150),(23,105,61,149),(24,104,62,148),(25,137,86,171),(26,136,87,170),(27,135,88,169),(28,134,89,192),(29,133,90,191),(30,132,91,190),(31,131,92,189),(32,130,93,188),(33,129,94,187),(34,128,95,186),(35,127,96,185),(36,126,73,184),(37,125,74,183),(38,124,75,182),(39,123,76,181),(40,122,77,180),(41,121,78,179),(42,144,79,178),(43,143,80,177),(44,142,81,176),(45,141,82,175),(46,140,83,174),(47,139,84,173),(48,138,85,172)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 6A | 6B | 6C | 8A | ··· | 8H | 12A | ··· | 12L | 24A | ··· | 24P |
| order | 1 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | ··· | 8 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | ··· | 2 | 24 | 24 | 24 | 24 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | + | + | + | - | - | + | + | - |
| image | C1 | C2 | C2 | C2 | S3 | Q8 | D4 | D6 | D6 | D8 | Q16 | Dic6 | D12 | D24 | Dic12 |
| kernel | C24⋊8Q8 | C24⋊1C4 | C4×C24 | C12⋊2Q8 | C4×C8 | C24 | C2×C12 | C42 | C2×C8 | C12 | C12 | C8 | C2×C4 | C4 | C4 |
| # reps | 1 | 4 | 1 | 2 | 1 | 4 | 2 | 1 | 2 | 4 | 4 | 8 | 4 | 8 | 8 |
Matrix representation of C24⋊8Q8 ►in GL4(𝔽73) generated by
| 0 | 72 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 5 | 50 |
| 0 | 0 | 23 | 55 |
| 0 | 1 | 0 | 0 |
| 72 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 28 | 23 | 0 | 0 |
| 23 | 45 | 0 | 0 |
| 0 | 0 | 25 | 19 |
| 0 | 0 | 44 | 48 |
G:=sub<GL(4,GF(73))| [0,1,0,0,72,0,0,0,0,0,5,23,0,0,50,55],[0,72,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[28,23,0,0,23,45,0,0,0,0,25,44,0,0,19,48] >;
C24⋊8Q8 in GAP, Magma, Sage, TeX
C_{24}\rtimes_8Q_8 % in TeX
G:=Group("C24:8Q8"); // GroupNames label
G:=SmallGroup(192,241);
// by ID
G=gap.SmallGroup(192,241);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,253,120,254,226,1123,136,6278]);
// Polycyclic
G:=Group<a,b,c|a^24=b^4=1,c^2=b^2,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^-1>;
// generators/relations