metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C6.1Q32, C24.78D4, C6.1SD32, Dic12⋊1C4, C2.1Dic24, C22.7D24, C12.22SD16, C8.18(C4×S3), (C2×C48).2C2, (C2×C16).2S3, (C2×C6).13D8, C24.48(C2×C4), (C2×C8).296D6, (C2×C4).70D12, C24⋊1C4.1C2, C4.1(C24⋊C2), C4.15(D6⋊C4), C3⋊2(C2.Q32), (C2×C12).369D4, C8.35(C3⋊D4), C2.1(C48⋊C2), (C2×Dic12).1C2, C2.6(C2.D24), C6.14(D4⋊C4), C12.39(C22⋊C4), (C2×C24).369C22, SmallGroup(192,62)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2.Dic24
G = < a,b,c | a2=b48=1, c2=ab24, ab=ba, ac=ca, cbc-1=ab-1 >
Subgroups: 200 in 58 conjugacy classes, 29 normal (27 characteristic)
C1, C2, C3, C4, C4, C22, C6, C8, C2×C4, C2×C4, Q8, Dic3, C12, C2×C6, C16, C4⋊C4, C2×C8, Q16, C2×Q8, C24, Dic6, C2×Dic3, C2×C12, C2.D8, C2×C16, C2×Q16, C48, Dic12, Dic12, C4⋊Dic3, C2×C24, C2×Dic6, C2.Q32, C24⋊1C4, C2×C48, C2×Dic12, C2.Dic24
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D6, C22⋊C4, D8, SD16, C4×S3, D12, C3⋊D4, D4⋊C4, SD32, Q32, C24⋊C2, D24, D6⋊C4, C2.Q32, C48⋊C2, Dic24, C2.D24, C2.Dic24
►Regular action on 192 points
Generators in S192
(1 157)(2 158)(3 159)(4 160)(5 161)(6 162)(7 163)(8 164)(9 165)(10 166)(11 167)(12 168)(13 169)(14 170)(15 171)(16 172)(17 173)(18 174)(19 175)(20 176)(21 177)(22 178)(23 179)(24 180)(25 181)(26 182)(27 183)(28 184)(29 185)(30 186)(31 187)(32 188)(33 189)(34 190)(35 191)(36 192)(37 145)(38 146)(39 147)(40 148)(41 149)(42 150)(43 151)(44 152)(45 153)(46 154)(47 155)(48 156)(49 143)(50 144)(51 97)(52 98)(53 99)(54 100)(55 101)(56 102)(57 103)(58 104)(59 105)(60 106)(61 107)(62 108)(63 109)(64 110)(65 111)(66 112)(67 113)(68 114)(69 115)(70 116)(71 117)(72 118)(73 119)(74 120)(75 121)(76 122)(77 123)(78 124)(79 125)(80 126)(81 127)(82 128)(83 129)(84 130)(85 131)(86 132)(87 133)(88 134)(89 135)(90 136)(91 137)(92 138)(93 139)(94 140)(95 141)(96 142)
(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 83 181 105)(2 128 182 58)(3 81 183 103)(4 126 184 56)(5 79 185 101)(6 124 186 54)(7 77 187 99)(8 122 188 52)(9 75 189 97)(10 120 190 50)(11 73 191 143)(12 118 192 96)(13 71 145 141)(14 116 146 94)(15 69 147 139)(16 114 148 92)(17 67 149 137)(18 112 150 90)(19 65 151 135)(20 110 152 88)(21 63 153 133)(22 108 154 86)(23 61 155 131)(24 106 156 84)(25 59 157 129)(26 104 158 82)(27 57 159 127)(28 102 160 80)(29 55 161 125)(30 100 162 78)(31 53 163 123)(32 98 164 76)(33 51 165 121)(34 144 166 74)(35 49 167 119)(36 142 168 72)(37 95 169 117)(38 140 170 70)(39 93 171 115)(40 138 172 68)(41 91 173 113)(42 136 174 66)(43 89 175 111)(44 134 176 64)(45 87 177 109)(46 132 178 62)(47 85 179 107)(48 130 180 60)
G:=sub<Sym(192)| (1,157)(2,158)(3,159)(4,160)(5,161)(6,162)(7,163)(8,164)(9,165)(10,166)(11,167)(12,168)(13,169)(14,170)(15,171)(16,172)(17,173)(18,174)(19,175)(20,176)(21,177)(22,178)(23,179)(24,180)(25,181)(26,182)(27,183)(28,184)(29,185)(30,186)(31,187)(32,188)(33,189)(34,190)(35,191)(36,192)(37,145)(38,146)(39,147)(40,148)(41,149)(42,150)(43,151)(44,152)(45,153)(46,154)(47,155)(48,156)(49,143)(50,144)(51,97)(52,98)(53,99)(54,100)(55,101)(56,102)(57,103)(58,104)(59,105)(60,106)(61,107)(62,108)(63,109)(64,110)(65,111)(66,112)(67,113)(68,114)(69,115)(70,116)(71,117)(72,118)(73,119)(74,120)(75,121)(76,122)(77,123)(78,124)(79,125)(80,126)(81,127)(82,128)(83,129)(84,130)(85,131)(86,132)(87,133)(88,134)(89,135)(90,136)(91,137)(92,138)(93,139)(94,140)(95,141)(96,142), (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), (1,83,181,105)(2,128,182,58)(3,81,183,103)(4,126,184,56)(5,79,185,101)(6,124,186,54)(7,77,187,99)(8,122,188,52)(9,75,189,97)(10,120,190,50)(11,73,191,143)(12,118,192,96)(13,71,145,141)(14,116,146,94)(15,69,147,139)(16,114,148,92)(17,67,149,137)(18,112,150,90)(19,65,151,135)(20,110,152,88)(21,63,153,133)(22,108,154,86)(23,61,155,131)(24,106,156,84)(25,59,157,129)(26,104,158,82)(27,57,159,127)(28,102,160,80)(29,55,161,125)(30,100,162,78)(31,53,163,123)(32,98,164,76)(33,51,165,121)(34,144,166,74)(35,49,167,119)(36,142,168,72)(37,95,169,117)(38,140,170,70)(39,93,171,115)(40,138,172,68)(41,91,173,113)(42,136,174,66)(43,89,175,111)(44,134,176,64)(45,87,177,109)(46,132,178,62)(47,85,179,107)(48,130,180,60)>;
G:=Group( (1,157)(2,158)(3,159)(4,160)(5,161)(6,162)(7,163)(8,164)(9,165)(10,166)(11,167)(12,168)(13,169)(14,170)(15,171)(16,172)(17,173)(18,174)(19,175)(20,176)(21,177)(22,178)(23,179)(24,180)(25,181)(26,182)(27,183)(28,184)(29,185)(30,186)(31,187)(32,188)(33,189)(34,190)(35,191)(36,192)(37,145)(38,146)(39,147)(40,148)(41,149)(42,150)(43,151)(44,152)(45,153)(46,154)(47,155)(48,156)(49,143)(50,144)(51,97)(52,98)(53,99)(54,100)(55,101)(56,102)(57,103)(58,104)(59,105)(60,106)(61,107)(62,108)(63,109)(64,110)(65,111)(66,112)(67,113)(68,114)(69,115)(70,116)(71,117)(72,118)(73,119)(74,120)(75,121)(76,122)(77,123)(78,124)(79,125)(80,126)(81,127)(82,128)(83,129)(84,130)(85,131)(86,132)(87,133)(88,134)(89,135)(90,136)(91,137)(92,138)(93,139)(94,140)(95,141)(96,142), (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), (1,83,181,105)(2,128,182,58)(3,81,183,103)(4,126,184,56)(5,79,185,101)(6,124,186,54)(7,77,187,99)(8,122,188,52)(9,75,189,97)(10,120,190,50)(11,73,191,143)(12,118,192,96)(13,71,145,141)(14,116,146,94)(15,69,147,139)(16,114,148,92)(17,67,149,137)(18,112,150,90)(19,65,151,135)(20,110,152,88)(21,63,153,133)(22,108,154,86)(23,61,155,131)(24,106,156,84)(25,59,157,129)(26,104,158,82)(27,57,159,127)(28,102,160,80)(29,55,161,125)(30,100,162,78)(31,53,163,123)(32,98,164,76)(33,51,165,121)(34,144,166,74)(35,49,167,119)(36,142,168,72)(37,95,169,117)(38,140,170,70)(39,93,171,115)(40,138,172,68)(41,91,173,113)(42,136,174,66)(43,89,175,111)(44,134,176,64)(45,87,177,109)(46,132,178,62)(47,85,179,107)(48,130,180,60) );
G=PermutationGroup([[(1,157),(2,158),(3,159),(4,160),(5,161),(6,162),(7,163),(8,164),(9,165),(10,166),(11,167),(12,168),(13,169),(14,170),(15,171),(16,172),(17,173),(18,174),(19,175),(20,176),(21,177),(22,178),(23,179),(24,180),(25,181),(26,182),(27,183),(28,184),(29,185),(30,186),(31,187),(32,188),(33,189),(34,190),(35,191),(36,192),(37,145),(38,146),(39,147),(40,148),(41,149),(42,150),(43,151),(44,152),(45,153),(46,154),(47,155),(48,156),(49,143),(50,144),(51,97),(52,98),(53,99),(54,100),(55,101),(56,102),(57,103),(58,104),(59,105),(60,106),(61,107),(62,108),(63,109),(64,110),(65,111),(66,112),(67,113),(68,114),(69,115),(70,116),(71,117),(72,118),(73,119),(74,120),(75,121),(76,122),(77,123),(78,124),(79,125),(80,126),(81,127),(82,128),(83,129),(84,130),(85,131),(86,132),(87,133),(88,134),(89,135),(90,136),(91,137),(92,138),(93,139),(94,140),(95,141),(96,142)], [(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)], [(1,83,181,105),(2,128,182,58),(3,81,183,103),(4,126,184,56),(5,79,185,101),(6,124,186,54),(7,77,187,99),(8,122,188,52),(9,75,189,97),(10,120,190,50),(11,73,191,143),(12,118,192,96),(13,71,145,141),(14,116,146,94),(15,69,147,139),(16,114,148,92),(17,67,149,137),(18,112,150,90),(19,65,151,135),(20,110,152,88),(21,63,153,133),(22,108,154,86),(23,61,155,131),(24,106,156,84),(25,59,157,129),(26,104,158,82),(27,57,159,127),(28,102,160,80),(29,55,161,125),(30,100,162,78),(31,53,163,123),(32,98,164,76),(33,51,165,121),(34,144,166,74),(35,49,167,119),(36,142,168,72),(37,95,169,117),(38,140,170,70),(39,93,171,115),(40,138,172,68),(41,91,173,113),(42,136,174,66),(43,89,175,111),(44,134,176,64),(45,87,177,109),(46,132,178,62),(47,85,179,107),(48,130,180,60)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 16A | ··· | 16H | 24A | ··· | 24H | 48A | ··· | 48P |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 16 | ··· | 16 | 24 | ··· | 24 | 48 | ··· | 48 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 24 | 24 | 24 | 24 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | - | |||||||
| image | C1 | C2 | C2 | C2 | C4 | S3 | D4 | D4 | D6 | SD16 | D8 | C4×S3 | C3⋊D4 | D12 | SD32 | Q32 | C24⋊C2 | D24 | C48⋊C2 | Dic24 |
| kernel | C2.Dic24 | C24⋊1C4 | C2×C48 | C2×Dic12 | Dic12 | C2×C16 | C24 | C2×C12 | C2×C8 | C12 | C2×C6 | C8 | C8 | C2×C4 | C6 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 |
Matrix representation of C2.Dic24 ►in GL3(𝔽97) generated by
| 96 | 0 | 0 |
| 0 | 96 | 0 |
| 0 | 0 | 96 |
| 75 | 0 | 0 |
| 0 | 2 | 54 |
| 0 | 43 | 45 |
| 75 | 0 | 0 |
| 0 | 36 | 93 |
| 0 | 57 | 61 |
G:=sub<GL(3,GF(97))| [96,0,0,0,96,0,0,0,96],[75,0,0,0,2,43,0,54,45],[75,0,0,0,36,57,0,93,61] >;
C2.Dic24 in GAP, Magma, Sage, TeX
C_2.{\rm Dic}_{24} % in TeX
G:=Group("C2.Dic24"); // GroupNames label
G:=SmallGroup(192,62);
// by ID
G=gap.SmallGroup(192,62);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,280,85,204,422,268,1684,102,6278]);
// Polycyclic
G:=Group<a,b,c|a^2=b^48=1,c^2=a*b^24,a*b=b*a,a*c=c*a,c*b*c^-1=a*b^-1>;
// generators/relations