direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C24⋊1C4, C23.62D12, C22.15D24, C22.6Dic12, (C2×C24)⋊9C4, C24⋊29(C2×C4), C8⋊7(C2×Dic3), (C2×C8)⋊5Dic3, (C2×C6).22D8, C6.15(C2×D8), C6⋊2(C2.D8), C2.2(C2×D24), (C2×C8).306D6, (C2×C4).94D12, C12.36(C4⋊C4), C12.74(C2×Q8), (C2×C6).10Q16, C6.10(C2×Q16), (C2×C12).56Q8, (C2×C12).387D4, (C22×C8).10S3, C4.40(C2×Dic6), C2.3(C2×Dic12), (C2×C4).49Dic6, (C22×C24).14C2, C4.17(C4⋊Dic3), (C22×C4).440D6, (C22×C6).136D4, C22.51(C2×D12), C12.170(C22×C4), (C2×C12).764C23, (C2×C24).379C22, C4.24(C22×Dic3), C4⋊Dic3.280C22, C22.22(C4⋊Dic3), (C22×C12).516C22, C3⋊3(C2×C2.D8), C6.45(C2×C4⋊C4), (C2×C6).50(C4⋊C4), (C2×C6).154(C2×D4), C2.11(C2×C4⋊Dic3), (C2×C12).299(C2×C4), (C2×C4⋊Dic3).23C2, (C2×C4).82(C2×Dic3), (C2×C4).711(C22×S3), SmallGroup(192,664)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C24⋊1C4
G = < a,b,c | a2=b24=c4=1, ab=ba, ac=ca, cbc-1=b-1 >
Subgroups: 312 in 130 conjugacy classes, 87 normal (25 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, C2×C4, C23, Dic3, C12, C12, C2×C6, C2×C6, C4⋊C4, C2×C8, C22×C4, C22×C4, C24, C2×Dic3, C2×C12, C2×C12, C22×C6, C2.D8, C2×C4⋊C4, C22×C8, C4⋊Dic3, C4⋊Dic3, C2×C24, C22×Dic3, C22×C12, C2×C2.D8, C24⋊1C4, C2×C4⋊Dic3, C22×C24, C2×C24⋊1C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, C23, Dic3, D6, C4⋊C4, D8, Q16, C22×C4, C2×D4, C2×Q8, Dic6, D12, C2×Dic3, C22×S3, C2.D8, C2×C4⋊C4, C2×D8, C2×Q16, D24, Dic12, C4⋊Dic3, C2×Dic6, C2×D12, C22×Dic3, C2×C2.D8, C24⋊1C4, C2×D24, C2×Dic12, C2×C4⋊Dic3, C2×C24⋊1C4
►Regular action on 192 points
Generators in S192
(1 121)(2 122)(3 123)(4 124)(5 125)(6 126)(7 127)(8 128)(9 129)(10 130)(11 131)(12 132)(13 133)(14 134)(15 135)(16 136)(17 137)(18 138)(19 139)(20 140)(21 141)(22 142)(23 143)(24 144)(25 105)(26 106)(27 107)(28 108)(29 109)(30 110)(31 111)(32 112)(33 113)(34 114)(35 115)(36 116)(37 117)(38 118)(39 119)(40 120)(41 97)(42 98)(43 99)(44 100)(45 101)(46 102)(47 103)(48 104)(49 190)(50 191)(51 192)(52 169)(53 170)(54 171)(55 172)(56 173)(57 174)(58 175)(59 176)(60 177)(61 178)(62 179)(63 180)(64 181)(65 182)(66 183)(67 184)(68 185)(69 186)(70 187)(71 188)(72 189)(73 157)(74 158)(75 159)(76 160)(77 161)(78 162)(79 163)(80 164)(81 165)(82 166)(83 167)(84 168)(85 145)(86 146)(87 147)(88 148)(89 149)(90 150)(91 151)(92 152)(93 153)(94 154)(95 155)(96 156)
(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)
(1 111 60 88)(2 110 61 87)(3 109 62 86)(4 108 63 85)(5 107 64 84)(6 106 65 83)(7 105 66 82)(8 104 67 81)(9 103 68 80)(10 102 69 79)(11 101 70 78)(12 100 71 77)(13 99 72 76)(14 98 49 75)(15 97 50 74)(16 120 51 73)(17 119 52 96)(18 118 53 95)(19 117 54 94)(20 116 55 93)(21 115 56 92)(22 114 57 91)(23 113 58 90)(24 112 59 89)(25 183 166 127)(26 182 167 126)(27 181 168 125)(28 180 145 124)(29 179 146 123)(30 178 147 122)(31 177 148 121)(32 176 149 144)(33 175 150 143)(34 174 151 142)(35 173 152 141)(36 172 153 140)(37 171 154 139)(38 170 155 138)(39 169 156 137)(40 192 157 136)(41 191 158 135)(42 190 159 134)(43 189 160 133)(44 188 161 132)(45 187 162 131)(46 186 163 130)(47 185 164 129)(48 184 165 128)
G:=sub<Sym(192)| (1,121)(2,122)(3,123)(4,124)(5,125)(6,126)(7,127)(8,128)(9,129)(10,130)(11,131)(12,132)(13,133)(14,134)(15,135)(16,136)(17,137)(18,138)(19,139)(20,140)(21,141)(22,142)(23,143)(24,144)(25,105)(26,106)(27,107)(28,108)(29,109)(30,110)(31,111)(32,112)(33,113)(34,114)(35,115)(36,116)(37,117)(38,118)(39,119)(40,120)(41,97)(42,98)(43,99)(44,100)(45,101)(46,102)(47,103)(48,104)(49,190)(50,191)(51,192)(52,169)(53,170)(54,171)(55,172)(56,173)(57,174)(58,175)(59,176)(60,177)(61,178)(62,179)(63,180)(64,181)(65,182)(66,183)(67,184)(68,185)(69,186)(70,187)(71,188)(72,189)(73,157)(74,158)(75,159)(76,160)(77,161)(78,162)(79,163)(80,164)(81,165)(82,166)(83,167)(84,168)(85,145)(86,146)(87,147)(88,148)(89,149)(90,150)(91,151)(92,152)(93,153)(94,154)(95,155)(96,156), (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), (1,111,60,88)(2,110,61,87)(3,109,62,86)(4,108,63,85)(5,107,64,84)(6,106,65,83)(7,105,66,82)(8,104,67,81)(9,103,68,80)(10,102,69,79)(11,101,70,78)(12,100,71,77)(13,99,72,76)(14,98,49,75)(15,97,50,74)(16,120,51,73)(17,119,52,96)(18,118,53,95)(19,117,54,94)(20,116,55,93)(21,115,56,92)(22,114,57,91)(23,113,58,90)(24,112,59,89)(25,183,166,127)(26,182,167,126)(27,181,168,125)(28,180,145,124)(29,179,146,123)(30,178,147,122)(31,177,148,121)(32,176,149,144)(33,175,150,143)(34,174,151,142)(35,173,152,141)(36,172,153,140)(37,171,154,139)(38,170,155,138)(39,169,156,137)(40,192,157,136)(41,191,158,135)(42,190,159,134)(43,189,160,133)(44,188,161,132)(45,187,162,131)(46,186,163,130)(47,185,164,129)(48,184,165,128)>;
G:=Group( (1,121)(2,122)(3,123)(4,124)(5,125)(6,126)(7,127)(8,128)(9,129)(10,130)(11,131)(12,132)(13,133)(14,134)(15,135)(16,136)(17,137)(18,138)(19,139)(20,140)(21,141)(22,142)(23,143)(24,144)(25,105)(26,106)(27,107)(28,108)(29,109)(30,110)(31,111)(32,112)(33,113)(34,114)(35,115)(36,116)(37,117)(38,118)(39,119)(40,120)(41,97)(42,98)(43,99)(44,100)(45,101)(46,102)(47,103)(48,104)(49,190)(50,191)(51,192)(52,169)(53,170)(54,171)(55,172)(56,173)(57,174)(58,175)(59,176)(60,177)(61,178)(62,179)(63,180)(64,181)(65,182)(66,183)(67,184)(68,185)(69,186)(70,187)(71,188)(72,189)(73,157)(74,158)(75,159)(76,160)(77,161)(78,162)(79,163)(80,164)(81,165)(82,166)(83,167)(84,168)(85,145)(86,146)(87,147)(88,148)(89,149)(90,150)(91,151)(92,152)(93,153)(94,154)(95,155)(96,156), (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), (1,111,60,88)(2,110,61,87)(3,109,62,86)(4,108,63,85)(5,107,64,84)(6,106,65,83)(7,105,66,82)(8,104,67,81)(9,103,68,80)(10,102,69,79)(11,101,70,78)(12,100,71,77)(13,99,72,76)(14,98,49,75)(15,97,50,74)(16,120,51,73)(17,119,52,96)(18,118,53,95)(19,117,54,94)(20,116,55,93)(21,115,56,92)(22,114,57,91)(23,113,58,90)(24,112,59,89)(25,183,166,127)(26,182,167,126)(27,181,168,125)(28,180,145,124)(29,179,146,123)(30,178,147,122)(31,177,148,121)(32,176,149,144)(33,175,150,143)(34,174,151,142)(35,173,152,141)(36,172,153,140)(37,171,154,139)(38,170,155,138)(39,169,156,137)(40,192,157,136)(41,191,158,135)(42,190,159,134)(43,189,160,133)(44,188,161,132)(45,187,162,131)(46,186,163,130)(47,185,164,129)(48,184,165,128) );
G=PermutationGroup([[(1,121),(2,122),(3,123),(4,124),(5,125),(6,126),(7,127),(8,128),(9,129),(10,130),(11,131),(12,132),(13,133),(14,134),(15,135),(16,136),(17,137),(18,138),(19,139),(20,140),(21,141),(22,142),(23,143),(24,144),(25,105),(26,106),(27,107),(28,108),(29,109),(30,110),(31,111),(32,112),(33,113),(34,114),(35,115),(36,116),(37,117),(38,118),(39,119),(40,120),(41,97),(42,98),(43,99),(44,100),(45,101),(46,102),(47,103),(48,104),(49,190),(50,191),(51,192),(52,169),(53,170),(54,171),(55,172),(56,173),(57,174),(58,175),(59,176),(60,177),(61,178),(62,179),(63,180),(64,181),(65,182),(66,183),(67,184),(68,185),(69,186),(70,187),(71,188),(72,189),(73,157),(74,158),(75,159),(76,160),(77,161),(78,162),(79,163),(80,164),(81,165),(82,166),(83,167),(84,168),(85,145),(86,146),(87,147),(88,148),(89,149),(90,150),(91,151),(92,152),(93,153),(94,154),(95,155),(96,156)], [(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)], [(1,111,60,88),(2,110,61,87),(3,109,62,86),(4,108,63,85),(5,107,64,84),(6,106,65,83),(7,105,66,82),(8,104,67,81),(9,103,68,80),(10,102,69,79),(11,101,70,78),(12,100,71,77),(13,99,72,76),(14,98,49,75),(15,97,50,74),(16,120,51,73),(17,119,52,96),(18,118,53,95),(19,117,54,94),(20,116,55,93),(21,115,56,92),(22,114,57,91),(23,113,58,90),(24,112,59,89),(25,183,166,127),(26,182,167,126),(27,181,168,125),(28,180,145,124),(29,179,146,123),(30,178,147,122),(31,177,148,121),(32,176,149,144),(33,175,150,143),(34,174,151,142),(35,173,152,141),(36,172,153,140),(37,171,154,139),(38,170,155,138),(39,169,156,137),(40,192,157,136),(41,191,158,135),(42,190,159,134),(43,189,160,133),(44,188,161,132),(45,187,162,131),(46,186,163,130),(47,185,164,129),(48,184,165,128)]])
60 conjugacy classes
| class | 1 | 2A | ··· | 2G | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 6A | ··· | 6G | 8A | ··· | 8H | 12A | ··· | 12H | 24A | ··· | 24P |
| order | 1 | 2 | ··· | 2 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 8 | ··· | 8 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | 12 | ··· | 12 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | + | - | + | + | + | - | - | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C4 | S3 | D4 | Q8 | D4 | Dic3 | D6 | D6 | D8 | Q16 | Dic6 | D12 | D12 | D24 | Dic12 |
| kernel | C2×C24⋊1C4 | C24⋊1C4 | C2×C4⋊Dic3 | C22×C24 | C2×C24 | C22×C8 | C2×C12 | C2×C12 | C22×C6 | C2×C8 | C2×C8 | C22×C4 | C2×C6 | C2×C6 | C2×C4 | C2×C4 | C23 | C22 | C22 |
| # reps | 1 | 4 | 2 | 1 | 8 | 1 | 1 | 2 | 1 | 4 | 2 | 1 | 4 | 4 | 4 | 2 | 2 | 8 | 8 |
Matrix representation of C2×C24⋊1C4 ►in GL4(𝔽73) generated by
| 1 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 55 | 23 |
| 0 | 0 | 50 | 5 |
| 27 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 28 | 3 |
| 0 | 0 | 31 | 45 |
G:=sub<GL(4,GF(73))| [1,0,0,0,0,72,0,0,0,0,1,0,0,0,0,1],[72,0,0,0,0,72,0,0,0,0,55,50,0,0,23,5],[27,0,0,0,0,1,0,0,0,0,28,31,0,0,3,45] >;
C2×C24⋊1C4 in GAP, Magma, Sage, TeX
C_2\times C_{24}\rtimes_1C_4 % in TeX
G:=Group("C2xC24:1C4"); // GroupNames label
G:=SmallGroup(192,664);
// by ID
G=gap.SmallGroup(192,664);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,422,268,1684,102,6278]);
// Polycyclic
G:=Group<a,b,c|a^2=b^24=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations