direct product, abelian, monomial, 2-elementary
Aliases: C23×C24, SmallGroup(192,1454)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — 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, C3, C4, C4, C22, C6, C6, C8, C2×C4, C23, C12, C12, C2×C6, C2×C8, C22×C4, C24, C24, C2×C12, C22×C6, C22×C8, C23×C4, C2×C24, C22×C12, C23×C6, C23×C8, C22×C24, C23×C12, C23×C24
Quotients: C1, C2, C3, C4, C22, C6, C8, C2×C4, C23, C12, C2×C6, C2×C8, C22×C4, C24, C24, C2×C12, C22×C6, C22×C8, C23×C4, C2×C24, C22×C12, C23×C6, C23×C8, C22×C24, C23×C12, C23×C24
►Regular action on 192 points
Generators in S192
(1 182)(2 183)(3 184)(4 185)(5 186)(6 187)(7 188)(8 189)(9 190)(10 191)(11 192)(12 169)(13 170)(14 171)(15 172)(16 173)(17 174)(18 175)(19 176)(20 177)(21 178)(22 179)(23 180)(24 181)(25 69)(26 70)(27 71)(28 72)(29 49)(30 50)(31 51)(32 52)(33 53)(34 54)(35 55)(36 56)(37 57)(38 58)(39 59)(40 60)(41 61)(42 62)(43 63)(44 64)(45 65)(46 66)(47 67)(48 68)(73 109)(74 110)(75 111)(76 112)(77 113)(78 114)(79 115)(80 116)(81 117)(82 118)(83 119)(84 120)(85 97)(86 98)(87 99)(88 100)(89 101)(90 102)(91 103)(92 104)(93 105)(94 106)(95 107)(96 108)(121 151)(122 152)(123 153)(124 154)(125 155)(126 156)(127 157)(128 158)(129 159)(130 160)(131 161)(132 162)(133 163)(134 164)(135 165)(136 166)(137 167)(138 168)(139 145)(140 146)(141 147)(142 148)(143 149)(144 150)
(1 97)(2 98)(3 99)(4 100)(5 101)(6 102)(7 103)(8 104)(9 105)(10 106)(11 107)(12 108)(13 109)(14 110)(15 111)(16 112)(17 113)(18 114)(19 115)(20 116)(21 117)(22 118)(23 119)(24 120)(25 138)(26 139)(27 140)(28 141)(29 142)(30 143)(31 144)(32 121)(33 122)(34 123)(35 124)(36 125)(37 126)(38 127)(39 128)(40 129)(41 130)(42 131)(43 132)(44 133)(45 134)(46 135)(47 136)(48 137)(49 148)(50 149)(51 150)(52 151)(53 152)(54 153)(55 154)(56 155)(57 156)(58 157)(59 158)(60 159)(61 160)(62 161)(63 162)(64 163)(65 164)(66 165)(67 166)(68 167)(69 168)(70 145)(71 146)(72 147)(73 170)(74 171)(75 172)(76 173)(77 174)(78 175)(79 176)(80 177)(81 178)(82 179)(83 180)(84 181)(85 182)(86 183)(87 184)(88 185)(89 186)(90 187)(91 188)(92 189)(93 190)(94 191)(95 192)(96 169)
(1 161)(2 162)(3 163)(4 164)(5 165)(6 166)(7 167)(8 168)(9 145)(10 146)(11 147)(12 148)(13 149)(14 150)(15 151)(16 152)(17 153)(18 154)(19 155)(20 156)(21 157)(22 158)(23 159)(24 160)(25 92)(26 93)(27 94)(28 95)(29 96)(30 73)(31 74)(32 75)(33 76)(34 77)(35 78)(36 79)(37 80)(38 81)(39 82)(40 83)(41 84)(42 85)(43 86)(44 87)(45 88)(46 89)(47 90)(48 91)(49 108)(50 109)(51 110)(52 111)(53 112)(54 113)(55 114)(56 115)(57 116)(58 117)(59 118)(60 119)(61 120)(62 97)(63 98)(64 99)(65 100)(66 101)(67 102)(68 103)(69 104)(70 105)(71 106)(72 107)(121 172)(122 173)(123 174)(124 175)(125 176)(126 177)(127 178)(128 179)(129 180)(130 181)(131 182)(132 183)(133 184)(134 185)(135 186)(136 187)(137 188)(138 189)(139 190)(140 191)(141 192)(142 169)(143 170)(144 171)
(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)| (1,182)(2,183)(3,184)(4,185)(5,186)(6,187)(7,188)(8,189)(9,190)(10,191)(11,192)(12,169)(13,170)(14,171)(15,172)(16,173)(17,174)(18,175)(19,176)(20,177)(21,178)(22,179)(23,180)(24,181)(25,69)(26,70)(27,71)(28,72)(29,49)(30,50)(31,51)(32,52)(33,53)(34,54)(35,55)(36,56)(37,57)(38,58)(39,59)(40,60)(41,61)(42,62)(43,63)(44,64)(45,65)(46,66)(47,67)(48,68)(73,109)(74,110)(75,111)(76,112)(77,113)(78,114)(79,115)(80,116)(81,117)(82,118)(83,119)(84,120)(85,97)(86,98)(87,99)(88,100)(89,101)(90,102)(91,103)(92,104)(93,105)(94,106)(95,107)(96,108)(121,151)(122,152)(123,153)(124,154)(125,155)(126,156)(127,157)(128,158)(129,159)(130,160)(131,161)(132,162)(133,163)(134,164)(135,165)(136,166)(137,167)(138,168)(139,145)(140,146)(141,147)(142,148)(143,149)(144,150), (1,97)(2,98)(3,99)(4,100)(5,101)(6,102)(7,103)(8,104)(9,105)(10,106)(11,107)(12,108)(13,109)(14,110)(15,111)(16,112)(17,113)(18,114)(19,115)(20,116)(21,117)(22,118)(23,119)(24,120)(25,138)(26,139)(27,140)(28,141)(29,142)(30,143)(31,144)(32,121)(33,122)(34,123)(35,124)(36,125)(37,126)(38,127)(39,128)(40,129)(41,130)(42,131)(43,132)(44,133)(45,134)(46,135)(47,136)(48,137)(49,148)(50,149)(51,150)(52,151)(53,152)(54,153)(55,154)(56,155)(57,156)(58,157)(59,158)(60,159)(61,160)(62,161)(63,162)(64,163)(65,164)(66,165)(67,166)(68,167)(69,168)(70,145)(71,146)(72,147)(73,170)(74,171)(75,172)(76,173)(77,174)(78,175)(79,176)(80,177)(81,178)(82,179)(83,180)(84,181)(85,182)(86,183)(87,184)(88,185)(89,186)(90,187)(91,188)(92,189)(93,190)(94,191)(95,192)(96,169), (1,161)(2,162)(3,163)(4,164)(5,165)(6,166)(7,167)(8,168)(9,145)(10,146)(11,147)(12,148)(13,149)(14,150)(15,151)(16,152)(17,153)(18,154)(19,155)(20,156)(21,157)(22,158)(23,159)(24,160)(25,92)(26,93)(27,94)(28,95)(29,96)(30,73)(31,74)(32,75)(33,76)(34,77)(35,78)(36,79)(37,80)(38,81)(39,82)(40,83)(41,84)(42,85)(43,86)(44,87)(45,88)(46,89)(47,90)(48,91)(49,108)(50,109)(51,110)(52,111)(53,112)(54,113)(55,114)(56,115)(57,116)(58,117)(59,118)(60,119)(61,120)(62,97)(63,98)(64,99)(65,100)(66,101)(67,102)(68,103)(69,104)(70,105)(71,106)(72,107)(121,172)(122,173)(123,174)(124,175)(125,176)(126,177)(127,178)(128,179)(129,180)(130,181)(131,182)(132,183)(133,184)(134,185)(135,186)(136,187)(137,188)(138,189)(139,190)(140,191)(141,192)(142,169)(143,170)(144,171), (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( (1,182)(2,183)(3,184)(4,185)(5,186)(6,187)(7,188)(8,189)(9,190)(10,191)(11,192)(12,169)(13,170)(14,171)(15,172)(16,173)(17,174)(18,175)(19,176)(20,177)(21,178)(22,179)(23,180)(24,181)(25,69)(26,70)(27,71)(28,72)(29,49)(30,50)(31,51)(32,52)(33,53)(34,54)(35,55)(36,56)(37,57)(38,58)(39,59)(40,60)(41,61)(42,62)(43,63)(44,64)(45,65)(46,66)(47,67)(48,68)(73,109)(74,110)(75,111)(76,112)(77,113)(78,114)(79,115)(80,116)(81,117)(82,118)(83,119)(84,120)(85,97)(86,98)(87,99)(88,100)(89,101)(90,102)(91,103)(92,104)(93,105)(94,106)(95,107)(96,108)(121,151)(122,152)(123,153)(124,154)(125,155)(126,156)(127,157)(128,158)(129,159)(130,160)(131,161)(132,162)(133,163)(134,164)(135,165)(136,166)(137,167)(138,168)(139,145)(140,146)(141,147)(142,148)(143,149)(144,150), (1,97)(2,98)(3,99)(4,100)(5,101)(6,102)(7,103)(8,104)(9,105)(10,106)(11,107)(12,108)(13,109)(14,110)(15,111)(16,112)(17,113)(18,114)(19,115)(20,116)(21,117)(22,118)(23,119)(24,120)(25,138)(26,139)(27,140)(28,141)(29,142)(30,143)(31,144)(32,121)(33,122)(34,123)(35,124)(36,125)(37,126)(38,127)(39,128)(40,129)(41,130)(42,131)(43,132)(44,133)(45,134)(46,135)(47,136)(48,137)(49,148)(50,149)(51,150)(52,151)(53,152)(54,153)(55,154)(56,155)(57,156)(58,157)(59,158)(60,159)(61,160)(62,161)(63,162)(64,163)(65,164)(66,165)(67,166)(68,167)(69,168)(70,145)(71,146)(72,147)(73,170)(74,171)(75,172)(76,173)(77,174)(78,175)(79,176)(80,177)(81,178)(82,179)(83,180)(84,181)(85,182)(86,183)(87,184)(88,185)(89,186)(90,187)(91,188)(92,189)(93,190)(94,191)(95,192)(96,169), (1,161)(2,162)(3,163)(4,164)(5,165)(6,166)(7,167)(8,168)(9,145)(10,146)(11,147)(12,148)(13,149)(14,150)(15,151)(16,152)(17,153)(18,154)(19,155)(20,156)(21,157)(22,158)(23,159)(24,160)(25,92)(26,93)(27,94)(28,95)(29,96)(30,73)(31,74)(32,75)(33,76)(34,77)(35,78)(36,79)(37,80)(38,81)(39,82)(40,83)(41,84)(42,85)(43,86)(44,87)(45,88)(46,89)(47,90)(48,91)(49,108)(50,109)(51,110)(52,111)(53,112)(54,113)(55,114)(56,115)(57,116)(58,117)(59,118)(60,119)(61,120)(62,97)(63,98)(64,99)(65,100)(66,101)(67,102)(68,103)(69,104)(70,105)(71,106)(72,107)(121,172)(122,173)(123,174)(124,175)(125,176)(126,177)(127,178)(128,179)(129,180)(130,181)(131,182)(132,183)(133,184)(134,185)(135,186)(136,187)(137,188)(138,189)(139,190)(140,191)(141,192)(142,169)(143,170)(144,171), (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([[(1,182),(2,183),(3,184),(4,185),(5,186),(6,187),(7,188),(8,189),(9,190),(10,191),(11,192),(12,169),(13,170),(14,171),(15,172),(16,173),(17,174),(18,175),(19,176),(20,177),(21,178),(22,179),(23,180),(24,181),(25,69),(26,70),(27,71),(28,72),(29,49),(30,50),(31,51),(32,52),(33,53),(34,54),(35,55),(36,56),(37,57),(38,58),(39,59),(40,60),(41,61),(42,62),(43,63),(44,64),(45,65),(46,66),(47,67),(48,68),(73,109),(74,110),(75,111),(76,112),(77,113),(78,114),(79,115),(80,116),(81,117),(82,118),(83,119),(84,120),(85,97),(86,98),(87,99),(88,100),(89,101),(90,102),(91,103),(92,104),(93,105),(94,106),(95,107),(96,108),(121,151),(122,152),(123,153),(124,154),(125,155),(126,156),(127,157),(128,158),(129,159),(130,160),(131,161),(132,162),(133,163),(134,164),(135,165),(136,166),(137,167),(138,168),(139,145),(140,146),(141,147),(142,148),(143,149),(144,150)], [(1,97),(2,98),(3,99),(4,100),(5,101),(6,102),(7,103),(8,104),(9,105),(10,106),(11,107),(12,108),(13,109),(14,110),(15,111),(16,112),(17,113),(18,114),(19,115),(20,116),(21,117),(22,118),(23,119),(24,120),(25,138),(26,139),(27,140),(28,141),(29,142),(30,143),(31,144),(32,121),(33,122),(34,123),(35,124),(36,125),(37,126),(38,127),(39,128),(40,129),(41,130),(42,131),(43,132),(44,133),(45,134),(46,135),(47,136),(48,137),(49,148),(50,149),(51,150),(52,151),(53,152),(54,153),(55,154),(56,155),(57,156),(58,157),(59,158),(60,159),(61,160),(62,161),(63,162),(64,163),(65,164),(66,165),(67,166),(68,167),(69,168),(70,145),(71,146),(72,147),(73,170),(74,171),(75,172),(76,173),(77,174),(78,175),(79,176),(80,177),(81,178),(82,179),(83,180),(84,181),(85,182),(86,183),(87,184),(88,185),(89,186),(90,187),(91,188),(92,189),(93,190),(94,191),(95,192),(96,169)], [(1,161),(2,162),(3,163),(4,164),(5,165),(6,166),(7,167),(8,168),(9,145),(10,146),(11,147),(12,148),(13,149),(14,150),(15,151),(16,152),(17,153),(18,154),(19,155),(20,156),(21,157),(22,158),(23,159),(24,160),(25,92),(26,93),(27,94),(28,95),(29,96),(30,73),(31,74),(32,75),(33,76),(34,77),(35,78),(36,79),(37,80),(38,81),(39,82),(40,83),(41,84),(42,85),(43,86),(44,87),(45,88),(46,89),(47,90),(48,91),(49,108),(50,109),(51,110),(52,111),(53,112),(54,113),(55,114),(56,115),(57,116),(58,117),(59,118),(60,119),(61,120),(62,97),(63,98),(64,99),(65,100),(66,101),(67,102),(68,103),(69,104),(70,105),(71,106),(72,107),(121,172),(122,173),(123,174),(124,175),(125,176),(126,177),(127,178),(128,179),(129,180),(130,181),(131,182),(132,183),(133,184),(134,185),(135,186),(136,187),(137,188),(138,189),(139,190),(140,191),(141,192),(142,169),(143,170),(144,171)], [(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 | ··· | 2O | 3A | 3B | 4A | ··· | 4P | 6A | ··· | 6AD | 8A | ··· | 8AF | 12A | ··· | 12AF | 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 |
| type | + | + | + | |||||||||
| image | C1 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C8 | C12 | C12 | C24 |
| kernel | C23×C24 | C22×C24 | C23×C12 | C23×C8 | C22×C12 | C23×C6 | C22×C8 | C23×C4 | C22×C6 | C22×C4 | C24 | C23 |
| # reps | 1 | 14 | 1 | 2 | 14 | 2 | 28 | 2 | 32 | 28 | 4 | 64 |
Matrix representation of C23×C24 ►in GL4(𝔽73) generated by
| 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 |
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