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G = C2×C4×C24order 192 = 26·3

Abelian group of type [2,4,24]

direct product, abelian, monomial, 2-elementary

Aliases: C2×C4×C24, SmallGroup(192,835)

Series: Derived Chief Lower central Upper central

C1 — C2×C4×C24
C1C2C22C2×C4C2×C12C2×C24C4×C24 — C2×C4×C24
C1 — C2×C4×C24
C1 — C2×C4×C24

Generators and relations for C2×C4×C24
 G = < a,b,c | a2=b4=c24=1, ab=ba, ac=ca, bc=cb >

Subgroups: 162, all normal (18 characteristic)
C1, C2, C2 [×6], C3, C4 [×12], C22, C22 [×6], C6, C6 [×6], C8 [×8], C2×C4 [×2], C2×C4 [×16], C23, C12 [×12], C2×C6, C2×C6 [×6], C42 [×4], C2×C8 [×12], C22×C4, C22×C4 [×2], C24 [×8], C2×C12 [×2], C2×C12 [×16], C22×C6, C4×C8 [×4], C2×C42, C22×C8 [×2], C4×C12 [×4], C2×C24 [×12], C22×C12, C22×C12 [×2], C2×C4×C8, C4×C24 [×4], C2×C4×C12, C22×C24 [×2], C2×C4×C24
Quotients: C1, C2 [×7], C3, C4 [×12], C22 [×7], C6 [×7], C8 [×8], C2×C4 [×18], C23, C12 [×12], C2×C6 [×7], C42 [×4], C2×C8 [×12], C22×C4 [×3], C24 [×8], C2×C12 [×18], C22×C6, C4×C8 [×4], C2×C42, C22×C8 [×2], C4×C12 [×4], C2×C24 [×12], C22×C12 [×3], C2×C4×C8, C4×C24 [×4], C2×C4×C12, C22×C24 [×2], C2×C4×C24

Smallest permutation representation of C2×C4×C24
Regular action on 192 points
Generators in S192
(1 103)(2 104)(3 105)(4 106)(5 107)(6 108)(7 109)(8 110)(9 111)(10 112)(11 113)(12 114)(13 115)(14 116)(15 117)(16 118)(17 119)(18 120)(19 97)(20 98)(21 99)(22 100)(23 101)(24 102)(25 191)(26 192)(27 169)(28 170)(29 171)(30 172)(31 173)(32 174)(33 175)(34 176)(35 177)(36 178)(37 179)(38 180)(39 181)(40 182)(41 183)(42 184)(43 185)(44 186)(45 187)(46 188)(47 189)(48 190)(49 135)(50 136)(51 137)(52 138)(53 139)(54 140)(55 141)(56 142)(57 143)(58 144)(59 121)(60 122)(61 123)(62 124)(63 125)(64 126)(65 127)(66 128)(67 129)(68 130)(69 131)(70 132)(71 133)(72 134)(73 158)(74 159)(75 160)(76 161)(77 162)(78 163)(79 164)(80 165)(81 166)(82 167)(83 168)(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 64 46 85)(2 65 47 86)(3 66 48 87)(4 67 25 88)(5 68 26 89)(6 69 27 90)(7 70 28 91)(8 71 29 92)(9 72 30 93)(10 49 31 94)(11 50 32 95)(12 51 33 96)(13 52 34 73)(14 53 35 74)(15 54 36 75)(16 55 37 76)(17 56 38 77)(18 57 39 78)(19 58 40 79)(20 59 41 80)(21 60 42 81)(22 61 43 82)(23 62 44 83)(24 63 45 84)(97 144 182 164)(98 121 183 165)(99 122 184 166)(100 123 185 167)(101 124 186 168)(102 125 187 145)(103 126 188 146)(104 127 189 147)(105 128 190 148)(106 129 191 149)(107 130 192 150)(108 131 169 151)(109 132 170 152)(110 133 171 153)(111 134 172 154)(112 135 173 155)(113 136 174 156)(114 137 175 157)(115 138 176 158)(116 139 177 159)(117 140 178 160)(118 141 179 161)(119 142 180 162)(120 143 181 163)
(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)

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

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

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

192 conjugacy classes

class 1 2A···2G3A3B4A···4X6A···6N8A···8AF12A···12AV24A···24BL
order12···2334···46···68···812···1224···24
size11···1111···11···11···11···11···1

192 irreducible representations

dim1111111111111111
type++++
imageC1C2C2C2C3C4C4C4C6C6C6C8C12C12C12C24
kernelC2×C4×C24C4×C24C2×C4×C12C22×C24C2×C4×C8C4×C12C2×C24C22×C12C4×C8C2×C42C22×C8C2×C12C42C2×C8C22×C4C2×C4
# reps14122416482432832864

Matrix representation of C2×C4×C24 in GL3(𝔽73) generated by

7200
010
0072
,
7200
0270
001
,
7000
0270
0017
G:=sub<GL(3,GF(73))| [72,0,0,0,1,0,0,0,72],[72,0,0,0,27,0,0,0,1],[70,0,0,0,27,0,0,0,17] >;

C2×C4×C24 in GAP, Magma, Sage, TeX

C_2\times C_4\times C_{24}
% in TeX

G:=Group("C2xC4xC24");
// GroupNames label

G:=SmallGroup(192,835);
// by ID

G=gap.SmallGroup(192,835);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,168,344,172]);
// Polycyclic

G:=Group<a,b,c|a^2=b^4=c^24=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations

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