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## G = C2×C4×C24order 192 = 26·3

### Abelian group of type [2,4,24]

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C4×C24
 Chief series C1 — C2 — C22 — C2×C4 — C2×C12 — C2×C24 — C4×C24 — C2×C4×C24
 Lower central C1 — C2×C4×C24
 Upper central 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, C3, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, C23, C12, C2×C6, C2×C6, C42, C2×C8, C22×C4, C22×C4, C24, C2×C12, C2×C12, C22×C6, C4×C8, C2×C42, C22×C8, C4×C12, C2×C24, C22×C12, C22×C12, C2×C4×C8, C4×C24, C2×C4×C12, C22×C24, C2×C4×C24
Quotients: C1, C2, C3, C4, C22, C6, C8, C2×C4, C23, C12, C2×C6, C42, C2×C8, C22×C4, C24, C2×C12, C22×C6, C4×C8, C2×C42, C22×C8, C4×C12, C2×C24, C22×C12, C2×C4×C8, C4×C24, C2×C4×C12, C22×C24, C2×C4×C24

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

192 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 type + + + + image C1 C2 C2 C2 C3 C4 C4 C4 C6 C6 C6 C8 C12 C12 C12 C24 kernel C2×C4×C24 C4×C24 C2×C4×C12 C22×C24 C2×C4×C8 C4×C12 C2×C24 C22×C12 C4×C8 C2×C42 C22×C8 C2×C12 C42 C2×C8 C22×C4 C2×C4 # reps 1 4 1 2 2 4 16 4 8 2 4 32 8 32 8 64

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

 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
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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