Copied to
clipboard

## G = C2×Dic24order 192 = 26·3

### Direct product of C2 and Dic24

Series: Derived Chief Lower central Upper central

 Derived series C1 — C24 — C2×Dic24
 Chief series C1 — C3 — C6 — C12 — C24 — Dic12 — C2×Dic12 — C2×Dic24
 Lower central C3 — C6 — C12 — C24 — C2×Dic24
 Upper central C1 — C22 — C2×C4 — C2×C8 — C2×C16

Generators and relations for C2×Dic24
G = < a,b,c | a2=b48=1, c2=b24, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 264 in 82 conjugacy classes, 39 normal (21 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C6, C8, C2×C4, C2×C4, Q8, Dic3, C12, C2×C6, C16, C2×C8, Q16, C2×Q8, C24, Dic6, C2×Dic3, C2×C12, C2×C16, Q32, C2×Q16, C48, Dic12, Dic12, C2×C24, C2×Dic6, C2×Q32, Dic24, C2×C48, C2×Dic12, C2×Dic24
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, C2×D4, D12, C22×S3, Q32, C2×D8, D24, C2×D12, C2×Q32, Dic24, C2×D24, C2×Dic24

Smallest permutation representation of C2×Dic24
Regular action on 192 points
Generators in S192
(1 102)(2 103)(3 104)(4 105)(5 106)(6 107)(7 108)(8 109)(9 110)(10 111)(11 112)(12 113)(13 114)(14 115)(15 116)(16 117)(17 118)(18 119)(19 120)(20 121)(21 122)(22 123)(23 124)(24 125)(25 126)(26 127)(27 128)(28 129)(29 130)(30 131)(31 132)(32 133)(33 134)(34 135)(35 136)(36 137)(37 138)(38 139)(39 140)(40 141)(41 142)(42 143)(43 144)(44 97)(45 98)(46 99)(47 100)(48 101)(49 158)(50 159)(51 160)(52 161)(53 162)(54 163)(55 164)(56 165)(57 166)(58 167)(59 168)(60 169)(61 170)(62 171)(63 172)(64 173)(65 174)(66 175)(67 176)(68 177)(69 178)(70 179)(71 180)(72 181)(73 182)(74 183)(75 184)(76 185)(77 186)(78 187)(79 188)(80 189)(81 190)(82 191)(83 192)(84 145)(85 146)(86 147)(87 148)(88 149)(89 150)(90 151)(91 152)(92 153)(93 154)(94 155)(95 156)(96 157)
(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)
(1 187 25 163)(2 186 26 162)(3 185 27 161)(4 184 28 160)(5 183 29 159)(6 182 30 158)(7 181 31 157)(8 180 32 156)(9 179 33 155)(10 178 34 154)(11 177 35 153)(12 176 36 152)(13 175 37 151)(14 174 38 150)(15 173 39 149)(16 172 40 148)(17 171 41 147)(18 170 42 146)(19 169 43 145)(20 168 44 192)(21 167 45 191)(22 166 46 190)(23 165 47 189)(24 164 48 188)(49 107 73 131)(50 106 74 130)(51 105 75 129)(52 104 76 128)(53 103 77 127)(54 102 78 126)(55 101 79 125)(56 100 80 124)(57 99 81 123)(58 98 82 122)(59 97 83 121)(60 144 84 120)(61 143 85 119)(62 142 86 118)(63 141 87 117)(64 140 88 116)(65 139 89 115)(66 138 90 114)(67 137 91 113)(68 136 92 112)(69 135 93 111)(70 134 94 110)(71 133 95 109)(72 132 96 108)

G:=sub<Sym(192)| (1,102)(2,103)(3,104)(4,105)(5,106)(6,107)(7,108)(8,109)(9,110)(10,111)(11,112)(12,113)(13,114)(14,115)(15,116)(16,117)(17,118)(18,119)(19,120)(20,121)(21,122)(22,123)(23,124)(24,125)(25,126)(26,127)(27,128)(28,129)(29,130)(30,131)(31,132)(32,133)(33,134)(34,135)(35,136)(36,137)(37,138)(38,139)(39,140)(40,141)(41,142)(42,143)(43,144)(44,97)(45,98)(46,99)(47,100)(48,101)(49,158)(50,159)(51,160)(52,161)(53,162)(54,163)(55,164)(56,165)(57,166)(58,167)(59,168)(60,169)(61,170)(62,171)(63,172)(64,173)(65,174)(66,175)(67,176)(68,177)(69,178)(70,179)(71,180)(72,181)(73,182)(74,183)(75,184)(76,185)(77,186)(78,187)(79,188)(80,189)(81,190)(82,191)(83,192)(84,145)(85,146)(86,147)(87,148)(88,149)(89,150)(90,151)(91,152)(92,153)(93,154)(94,155)(95,156)(96,157), (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), (1,187,25,163)(2,186,26,162)(3,185,27,161)(4,184,28,160)(5,183,29,159)(6,182,30,158)(7,181,31,157)(8,180,32,156)(9,179,33,155)(10,178,34,154)(11,177,35,153)(12,176,36,152)(13,175,37,151)(14,174,38,150)(15,173,39,149)(16,172,40,148)(17,171,41,147)(18,170,42,146)(19,169,43,145)(20,168,44,192)(21,167,45,191)(22,166,46,190)(23,165,47,189)(24,164,48,188)(49,107,73,131)(50,106,74,130)(51,105,75,129)(52,104,76,128)(53,103,77,127)(54,102,78,126)(55,101,79,125)(56,100,80,124)(57,99,81,123)(58,98,82,122)(59,97,83,121)(60,144,84,120)(61,143,85,119)(62,142,86,118)(63,141,87,117)(64,140,88,116)(65,139,89,115)(66,138,90,114)(67,137,91,113)(68,136,92,112)(69,135,93,111)(70,134,94,110)(71,133,95,109)(72,132,96,108)>;

G:=Group( (1,102)(2,103)(3,104)(4,105)(5,106)(6,107)(7,108)(8,109)(9,110)(10,111)(11,112)(12,113)(13,114)(14,115)(15,116)(16,117)(17,118)(18,119)(19,120)(20,121)(21,122)(22,123)(23,124)(24,125)(25,126)(26,127)(27,128)(28,129)(29,130)(30,131)(31,132)(32,133)(33,134)(34,135)(35,136)(36,137)(37,138)(38,139)(39,140)(40,141)(41,142)(42,143)(43,144)(44,97)(45,98)(46,99)(47,100)(48,101)(49,158)(50,159)(51,160)(52,161)(53,162)(54,163)(55,164)(56,165)(57,166)(58,167)(59,168)(60,169)(61,170)(62,171)(63,172)(64,173)(65,174)(66,175)(67,176)(68,177)(69,178)(70,179)(71,180)(72,181)(73,182)(74,183)(75,184)(76,185)(77,186)(78,187)(79,188)(80,189)(81,190)(82,191)(83,192)(84,145)(85,146)(86,147)(87,148)(88,149)(89,150)(90,151)(91,152)(92,153)(93,154)(94,155)(95,156)(96,157), (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), (1,187,25,163)(2,186,26,162)(3,185,27,161)(4,184,28,160)(5,183,29,159)(6,182,30,158)(7,181,31,157)(8,180,32,156)(9,179,33,155)(10,178,34,154)(11,177,35,153)(12,176,36,152)(13,175,37,151)(14,174,38,150)(15,173,39,149)(16,172,40,148)(17,171,41,147)(18,170,42,146)(19,169,43,145)(20,168,44,192)(21,167,45,191)(22,166,46,190)(23,165,47,189)(24,164,48,188)(49,107,73,131)(50,106,74,130)(51,105,75,129)(52,104,76,128)(53,103,77,127)(54,102,78,126)(55,101,79,125)(56,100,80,124)(57,99,81,123)(58,98,82,122)(59,97,83,121)(60,144,84,120)(61,143,85,119)(62,142,86,118)(63,141,87,117)(64,140,88,116)(65,139,89,115)(66,138,90,114)(67,137,91,113)(68,136,92,112)(69,135,93,111)(70,134,94,110)(71,133,95,109)(72,132,96,108) );

G=PermutationGroup([[(1,102),(2,103),(3,104),(4,105),(5,106),(6,107),(7,108),(8,109),(9,110),(10,111),(11,112),(12,113),(13,114),(14,115),(15,116),(16,117),(17,118),(18,119),(19,120),(20,121),(21,122),(22,123),(23,124),(24,125),(25,126),(26,127),(27,128),(28,129),(29,130),(30,131),(31,132),(32,133),(33,134),(34,135),(35,136),(36,137),(37,138),(38,139),(39,140),(40,141),(41,142),(42,143),(43,144),(44,97),(45,98),(46,99),(47,100),(48,101),(49,158),(50,159),(51,160),(52,161),(53,162),(54,163),(55,164),(56,165),(57,166),(58,167),(59,168),(60,169),(61,170),(62,171),(63,172),(64,173),(65,174),(66,175),(67,176),(68,177),(69,178),(70,179),(71,180),(72,181),(73,182),(74,183),(75,184),(76,185),(77,186),(78,187),(79,188),(80,189),(81,190),(82,191),(83,192),(84,145),(85,146),(86,147),(87,148),(88,149),(89,150),(90,151),(91,152),(92,153),(93,154),(94,155),(95,156),(96,157)], [(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)], [(1,187,25,163),(2,186,26,162),(3,185,27,161),(4,184,28,160),(5,183,29,159),(6,182,30,158),(7,181,31,157),(8,180,32,156),(9,179,33,155),(10,178,34,154),(11,177,35,153),(12,176,36,152),(13,175,37,151),(14,174,38,150),(15,173,39,149),(16,172,40,148),(17,171,41,147),(18,170,42,146),(19,169,43,145),(20,168,44,192),(21,167,45,191),(22,166,46,190),(23,165,47,189),(24,164,48,188),(49,107,73,131),(50,106,74,130),(51,105,75,129),(52,104,76,128),(53,103,77,127),(54,102,78,126),(55,101,79,125),(56,100,80,124),(57,99,81,123),(58,98,82,122),(59,97,83,121),(60,144,84,120),(61,143,85,119),(62,142,86,118),(63,141,87,117),(64,140,88,116),(65,139,89,115),(66,138,90,114),(67,137,91,113),(68,136,92,112),(69,135,93,111),(70,134,94,110),(71,133,95,109),(72,132,96,108)]])

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 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + + + - + + - image C1 C2 C2 C2 S3 D4 D4 D6 D6 D8 D8 D12 D12 Q32 D24 D24 Dic24 kernel C2×Dic24 Dic24 C2×C48 C2×Dic12 C2×C16 C24 C2×C12 C16 C2×C8 C12 C2×C6 C8 C2×C4 C6 C4 C22 C2 # reps 1 4 1 2 1 1 1 2 1 2 2 2 2 8 4 4 16

Matrix representation of C2×Dic24 in GL3(𝔽97) generated by

 96 0 0 0 96 0 0 0 96
,
 96 0 0 0 73 77 0 20 53
,
 96 0 0 0 45 54 0 2 52
G:=sub<GL(3,GF(97))| [96,0,0,0,96,0,0,0,96],[96,0,0,0,73,20,0,77,53],[96,0,0,0,45,2,0,54,52] >;

C2×Dic24 in GAP, Magma, Sage, TeX

C_2\times {\rm Dic}_{24}
% in TeX

G:=Group("C2xDic24");
// GroupNames label

G:=SmallGroup(192,464);
// by ID

G=gap.SmallGroup(192,464);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,254,142,675,192,1684,102,6278]);
// Polycyclic

G:=Group<a,b,c|a^2=b^48=1,c^2=b^24,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

׿
×
𝔽