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G = C23×C24order 192 = 26·3

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

direct product, abelian, monomial, 2-elementary

Aliases: C23×C24, SmallGroup(192,1454)

Series: Derived Chief Lower central Upper central

C1 — C23×C24
C1C2C4C12C24C2×C24C22×C24 — C23×C24
C1 — C23×C24
C1 — C23×C24

Generators and relations for C23×C24
 G = < a,b,c,d | a2=b2=c2=d24=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, cd=dc >

Subgroups: 338, all normal (12 characteristic)
C1, C2, C2 [×14], C3, C4, C4 [×7], C22 [×35], C6, C6 [×14], C8 [×8], C2×C4 [×28], C23 [×15], C12, C12 [×7], C2×C6 [×35], C2×C8 [×28], C22×C4 [×14], C24, C24 [×8], C2×C12 [×28], C22×C6 [×15], C22×C8 [×14], C23×C4, C2×C24 [×28], C22×C12 [×14], C23×C6, C23×C8, C22×C24 [×14], C23×C12, C23×C24
Quotients: C1, C2 [×15], C3, C4 [×8], C22 [×35], C6 [×15], C8 [×8], C2×C4 [×28], C23 [×15], C12 [×8], C2×C6 [×35], C2×C8 [×28], C22×C4 [×14], C24, C24 [×8], C2×C12 [×28], C22×C6 [×15], C22×C8 [×14], C23×C4, C2×C24 [×28], C22×C12 [×14], C23×C6, C23×C8, C22×C24 [×14], C23×C12, C23×C24

Smallest permutation representation of C23×C24
Regular action on 192 points
Generators in S192
(1 111)(2 112)(3 113)(4 114)(5 115)(6 116)(7 117)(8 118)(9 119)(10 120)(11 97)(12 98)(13 99)(14 100)(15 101)(16 102)(17 103)(18 104)(19 105)(20 106)(21 107)(22 108)(23 109)(24 110)(25 77)(26 78)(27 79)(28 80)(29 81)(30 82)(31 83)(32 84)(33 85)(34 86)(35 87)(36 88)(37 89)(38 90)(39 91)(40 92)(41 93)(42 94)(43 95)(44 96)(45 73)(46 74)(47 75)(48 76)(49 173)(50 174)(51 175)(52 176)(53 177)(54 178)(55 179)(56 180)(57 181)(58 182)(59 183)(60 184)(61 185)(62 186)(63 187)(64 188)(65 189)(66 190)(67 191)(68 192)(69 169)(70 170)(71 171)(72 172)(121 168)(122 145)(123 146)(124 147)(125 148)(126 149)(127 150)(128 151)(129 152)(130 153)(131 154)(132 155)(133 156)(134 157)(135 158)(136 159)(137 160)(138 161)(139 162)(140 163)(141 164)(142 165)(143 166)(144 167)
(1 181)(2 182)(3 183)(4 184)(5 185)(6 186)(7 187)(8 188)(9 189)(10 190)(11 191)(12 192)(13 169)(14 170)(15 171)(16 172)(17 173)(18 174)(19 175)(20 176)(21 177)(22 178)(23 179)(24 180)(25 149)(26 150)(27 151)(28 152)(29 153)(30 154)(31 155)(32 156)(33 157)(34 158)(35 159)(36 160)(37 161)(38 162)(39 163)(40 164)(41 165)(42 166)(43 167)(44 168)(45 145)(46 146)(47 147)(48 148)(49 103)(50 104)(51 105)(52 106)(53 107)(54 108)(55 109)(56 110)(57 111)(58 112)(59 113)(60 114)(61 115)(62 116)(63 117)(64 118)(65 119)(66 120)(67 97)(68 98)(69 99)(70 100)(71 101)(72 102)(73 122)(74 123)(75 124)(76 125)(77 126)(78 127)(79 128)(80 129)(81 130)(82 131)(83 132)(84 133)(85 134)(86 135)(87 136)(88 137)(89 138)(90 139)(91 140)(92 141)(93 142)(94 143)(95 144)(96 121)
(1 33)(2 34)(3 35)(4 36)(5 37)(6 38)(7 39)(8 40)(9 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 48)(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 32)(49 126)(50 127)(51 128)(52 129)(53 130)(54 131)(55 132)(56 133)(57 134)(58 135)(59 136)(60 137)(61 138)(62 139)(63 140)(64 141)(65 142)(66 143)(67 144)(68 121)(69 122)(70 123)(71 124)(72 125)(73 99)(74 100)(75 101)(76 102)(77 103)(78 104)(79 105)(80 106)(81 107)(82 108)(83 109)(84 110)(85 111)(86 112)(87 113)(88 114)(89 115)(90 116)(91 117)(92 118)(93 119)(94 120)(95 97)(96 98)(145 169)(146 170)(147 171)(148 172)(149 173)(150 174)(151 175)(152 176)(153 177)(154 178)(155 179)(156 180)(157 181)(158 182)(159 183)(160 184)(161 185)(162 186)(163 187)(164 188)(165 189)(166 190)(167 191)(168 192)
(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)| (1,111)(2,112)(3,113)(4,114)(5,115)(6,116)(7,117)(8,118)(9,119)(10,120)(11,97)(12,98)(13,99)(14,100)(15,101)(16,102)(17,103)(18,104)(19,105)(20,106)(21,107)(22,108)(23,109)(24,110)(25,77)(26,78)(27,79)(28,80)(29,81)(30,82)(31,83)(32,84)(33,85)(34,86)(35,87)(36,88)(37,89)(38,90)(39,91)(40,92)(41,93)(42,94)(43,95)(44,96)(45,73)(46,74)(47,75)(48,76)(49,173)(50,174)(51,175)(52,176)(53,177)(54,178)(55,179)(56,180)(57,181)(58,182)(59,183)(60,184)(61,185)(62,186)(63,187)(64,188)(65,189)(66,190)(67,191)(68,192)(69,169)(70,170)(71,171)(72,172)(121,168)(122,145)(123,146)(124,147)(125,148)(126,149)(127,150)(128,151)(129,152)(130,153)(131,154)(132,155)(133,156)(134,157)(135,158)(136,159)(137,160)(138,161)(139,162)(140,163)(141,164)(142,165)(143,166)(144,167), (1,181)(2,182)(3,183)(4,184)(5,185)(6,186)(7,187)(8,188)(9,189)(10,190)(11,191)(12,192)(13,169)(14,170)(15,171)(16,172)(17,173)(18,174)(19,175)(20,176)(21,177)(22,178)(23,179)(24,180)(25,149)(26,150)(27,151)(28,152)(29,153)(30,154)(31,155)(32,156)(33,157)(34,158)(35,159)(36,160)(37,161)(38,162)(39,163)(40,164)(41,165)(42,166)(43,167)(44,168)(45,145)(46,146)(47,147)(48,148)(49,103)(50,104)(51,105)(52,106)(53,107)(54,108)(55,109)(56,110)(57,111)(58,112)(59,113)(60,114)(61,115)(62,116)(63,117)(64,118)(65,119)(66,120)(67,97)(68,98)(69,99)(70,100)(71,101)(72,102)(73,122)(74,123)(75,124)(76,125)(77,126)(78,127)(79,128)(80,129)(81,130)(82,131)(83,132)(84,133)(85,134)(86,135)(87,136)(88,137)(89,138)(90,139)(91,140)(92,141)(93,142)(94,143)(95,144)(96,121), (1,33)(2,34)(3,35)(4,36)(5,37)(6,38)(7,39)(8,40)(9,41)(10,42)(11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32)(49,126)(50,127)(51,128)(52,129)(53,130)(54,131)(55,132)(56,133)(57,134)(58,135)(59,136)(60,137)(61,138)(62,139)(63,140)(64,141)(65,142)(66,143)(67,144)(68,121)(69,122)(70,123)(71,124)(72,125)(73,99)(74,100)(75,101)(76,102)(77,103)(78,104)(79,105)(80,106)(81,107)(82,108)(83,109)(84,110)(85,111)(86,112)(87,113)(88,114)(89,115)(90,116)(91,117)(92,118)(93,119)(94,120)(95,97)(96,98)(145,169)(146,170)(147,171)(148,172)(149,173)(150,174)(151,175)(152,176)(153,177)(154,178)(155,179)(156,180)(157,181)(158,182)(159,183)(160,184)(161,185)(162,186)(163,187)(164,188)(165,189)(166,190)(167,191)(168,192), (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( (1,111)(2,112)(3,113)(4,114)(5,115)(6,116)(7,117)(8,118)(9,119)(10,120)(11,97)(12,98)(13,99)(14,100)(15,101)(16,102)(17,103)(18,104)(19,105)(20,106)(21,107)(22,108)(23,109)(24,110)(25,77)(26,78)(27,79)(28,80)(29,81)(30,82)(31,83)(32,84)(33,85)(34,86)(35,87)(36,88)(37,89)(38,90)(39,91)(40,92)(41,93)(42,94)(43,95)(44,96)(45,73)(46,74)(47,75)(48,76)(49,173)(50,174)(51,175)(52,176)(53,177)(54,178)(55,179)(56,180)(57,181)(58,182)(59,183)(60,184)(61,185)(62,186)(63,187)(64,188)(65,189)(66,190)(67,191)(68,192)(69,169)(70,170)(71,171)(72,172)(121,168)(122,145)(123,146)(124,147)(125,148)(126,149)(127,150)(128,151)(129,152)(130,153)(131,154)(132,155)(133,156)(134,157)(135,158)(136,159)(137,160)(138,161)(139,162)(140,163)(141,164)(142,165)(143,166)(144,167), (1,181)(2,182)(3,183)(4,184)(5,185)(6,186)(7,187)(8,188)(9,189)(10,190)(11,191)(12,192)(13,169)(14,170)(15,171)(16,172)(17,173)(18,174)(19,175)(20,176)(21,177)(22,178)(23,179)(24,180)(25,149)(26,150)(27,151)(28,152)(29,153)(30,154)(31,155)(32,156)(33,157)(34,158)(35,159)(36,160)(37,161)(38,162)(39,163)(40,164)(41,165)(42,166)(43,167)(44,168)(45,145)(46,146)(47,147)(48,148)(49,103)(50,104)(51,105)(52,106)(53,107)(54,108)(55,109)(56,110)(57,111)(58,112)(59,113)(60,114)(61,115)(62,116)(63,117)(64,118)(65,119)(66,120)(67,97)(68,98)(69,99)(70,100)(71,101)(72,102)(73,122)(74,123)(75,124)(76,125)(77,126)(78,127)(79,128)(80,129)(81,130)(82,131)(83,132)(84,133)(85,134)(86,135)(87,136)(88,137)(89,138)(90,139)(91,140)(92,141)(93,142)(94,143)(95,144)(96,121), (1,33)(2,34)(3,35)(4,36)(5,37)(6,38)(7,39)(8,40)(9,41)(10,42)(11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32)(49,126)(50,127)(51,128)(52,129)(53,130)(54,131)(55,132)(56,133)(57,134)(58,135)(59,136)(60,137)(61,138)(62,139)(63,140)(64,141)(65,142)(66,143)(67,144)(68,121)(69,122)(70,123)(71,124)(72,125)(73,99)(74,100)(75,101)(76,102)(77,103)(78,104)(79,105)(80,106)(81,107)(82,108)(83,109)(84,110)(85,111)(86,112)(87,113)(88,114)(89,115)(90,116)(91,117)(92,118)(93,119)(94,120)(95,97)(96,98)(145,169)(146,170)(147,171)(148,172)(149,173)(150,174)(151,175)(152,176)(153,177)(154,178)(155,179)(156,180)(157,181)(158,182)(159,183)(160,184)(161,185)(162,186)(163,187)(164,188)(165,189)(166,190)(167,191)(168,192), (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([(1,111),(2,112),(3,113),(4,114),(5,115),(6,116),(7,117),(8,118),(9,119),(10,120),(11,97),(12,98),(13,99),(14,100),(15,101),(16,102),(17,103),(18,104),(19,105),(20,106),(21,107),(22,108),(23,109),(24,110),(25,77),(26,78),(27,79),(28,80),(29,81),(30,82),(31,83),(32,84),(33,85),(34,86),(35,87),(36,88),(37,89),(38,90),(39,91),(40,92),(41,93),(42,94),(43,95),(44,96),(45,73),(46,74),(47,75),(48,76),(49,173),(50,174),(51,175),(52,176),(53,177),(54,178),(55,179),(56,180),(57,181),(58,182),(59,183),(60,184),(61,185),(62,186),(63,187),(64,188),(65,189),(66,190),(67,191),(68,192),(69,169),(70,170),(71,171),(72,172),(121,168),(122,145),(123,146),(124,147),(125,148),(126,149),(127,150),(128,151),(129,152),(130,153),(131,154),(132,155),(133,156),(134,157),(135,158),(136,159),(137,160),(138,161),(139,162),(140,163),(141,164),(142,165),(143,166),(144,167)], [(1,181),(2,182),(3,183),(4,184),(5,185),(6,186),(7,187),(8,188),(9,189),(10,190),(11,191),(12,192),(13,169),(14,170),(15,171),(16,172),(17,173),(18,174),(19,175),(20,176),(21,177),(22,178),(23,179),(24,180),(25,149),(26,150),(27,151),(28,152),(29,153),(30,154),(31,155),(32,156),(33,157),(34,158),(35,159),(36,160),(37,161),(38,162),(39,163),(40,164),(41,165),(42,166),(43,167),(44,168),(45,145),(46,146),(47,147),(48,148),(49,103),(50,104),(51,105),(52,106),(53,107),(54,108),(55,109),(56,110),(57,111),(58,112),(59,113),(60,114),(61,115),(62,116),(63,117),(64,118),(65,119),(66,120),(67,97),(68,98),(69,99),(70,100),(71,101),(72,102),(73,122),(74,123),(75,124),(76,125),(77,126),(78,127),(79,128),(80,129),(81,130),(82,131),(83,132),(84,133),(85,134),(86,135),(87,136),(88,137),(89,138),(90,139),(91,140),(92,141),(93,142),(94,143),(95,144),(96,121)], [(1,33),(2,34),(3,35),(4,36),(5,37),(6,38),(7,39),(8,40),(9,41),(10,42),(11,43),(12,44),(13,45),(14,46),(15,47),(16,48),(17,25),(18,26),(19,27),(20,28),(21,29),(22,30),(23,31),(24,32),(49,126),(50,127),(51,128),(52,129),(53,130),(54,131),(55,132),(56,133),(57,134),(58,135),(59,136),(60,137),(61,138),(62,139),(63,140),(64,141),(65,142),(66,143),(67,144),(68,121),(69,122),(70,123),(71,124),(72,125),(73,99),(74,100),(75,101),(76,102),(77,103),(78,104),(79,105),(80,106),(81,107),(82,108),(83,109),(84,110),(85,111),(86,112),(87,113),(88,114),(89,115),(90,116),(91,117),(92,118),(93,119),(94,120),(95,97),(96,98),(145,169),(146,170),(147,171),(148,172),(149,173),(150,174),(151,175),(152,176),(153,177),(154,178),(155,179),(156,180),(157,181),(158,182),(159,183),(160,184),(161,185),(162,186),(163,187),(164,188),(165,189),(166,190),(167,191),(168,192)], [(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···2O3A3B4A···4P6A···6AD8A···8AF12A···12AF24A···24BL
order12···2334···46···68···812···1224···24
size11···1111···11···11···11···11···1

192 irreducible representations

dim111111111111
type+++
imageC1C2C2C3C4C4C6C6C8C12C12C24
kernelC23×C24C22×C24C23×C12C23×C8C22×C12C23×C6C22×C8C23×C4C22×C6C22×C4C24C23
# reps114121422823228464

Matrix representation of C23×C24 in GL4(𝔽73) generated by

1000
0100
0010
00072
,
1000
07200
00720
00072
,
72000
0100
00720
0001
,
52000
05200
00210
00063
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,72],[1,0,0,0,0,72,0,0,0,0,72,0,0,0,0,72],[72,0,0,0,0,1,0,0,0,0,72,0,0,0,0,1],[52,0,0,0,0,52,0,0,0,0,21,0,0,0,0,63] >;

C23×C24 in GAP, Magma, Sage, TeX

C_2^3\times C_{24}
% in TeX

G:=Group("C2^3xC24");
// GroupNames label

G:=SmallGroup(192,1454);
// by ID

G=gap.SmallGroup(192,1454);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-2,336,124]);
// Polycyclic

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

׿
×
𝔽