direct product, abelian, monomial, 2-elementary
Aliases: C23×C22, SmallGroup(176,42)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C23×C22 |
| C1 — C23×C22 |
| C1 — C23×C22 |
Generators and relations for C23×C22
G = < a,b,c,d | a2=b2=c2=d22=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, cd=dc >
Subgroups: 134, all normal (4 characteristic)
C1, C2, C22, C23, C11, C24, C22, C2×C22, C22×C22, C23×C22
Quotients: C1, C2, C22, C23, C11, C24, C22, C2×C22, C22×C22, C23×C22
►Regular action on 176 points
Generators in S176
(1 108)(2 109)(3 110)(4 89)(5 90)(6 91)(7 92)(8 93)(9 94)(10 95)(11 96)(12 97)(13 98)(14 99)(15 100)(16 101)(17 102)(18 103)(19 104)(20 105)(21 106)(22 107)(23 131)(24 132)(25 111)(26 112)(27 113)(28 114)(29 115)(30 116)(31 117)(32 118)(33 119)(34 120)(35 121)(36 122)(37 123)(38 124)(39 125)(40 126)(41 127)(42 128)(43 129)(44 130)(45 136)(46 137)(47 138)(48 139)(49 140)(50 141)(51 142)(52 143)(53 144)(54 145)(55 146)(56 147)(57 148)(58 149)(59 150)(60 151)(61 152)(62 153)(63 154)(64 133)(65 134)(66 135)(67 158)(68 159)(69 160)(70 161)(71 162)(72 163)(73 164)(74 165)(75 166)(76 167)(77 168)(78 169)(79 170)(80 171)(81 172)(82 173)(83 174)(84 175)(85 176)(86 155)(87 156)(88 157)
(1 63)(2 64)(3 65)(4 66)(5 45)(6 46)(7 47)(8 48)(9 49)(10 50)(11 51)(12 52)(13 53)(14 54)(15 55)(16 56)(17 57)(18 58)(19 59)(20 60)(21 61)(22 62)(23 84)(24 85)(25 86)(26 87)(27 88)(28 67)(29 68)(30 69)(31 70)(32 71)(33 72)(34 73)(35 74)(36 75)(37 76)(38 77)(39 78)(40 79)(41 80)(42 81)(43 82)(44 83)(89 135)(90 136)(91 137)(92 138)(93 139)(94 140)(95 141)(96 142)(97 143)(98 144)(99 145)(100 146)(101 147)(102 148)(103 149)(104 150)(105 151)(106 152)(107 153)(108 154)(109 133)(110 134)(111 155)(112 156)(113 157)(114 158)(115 159)(116 160)(117 161)(118 162)(119 163)(120 164)(121 165)(122 166)(123 167)(124 168)(125 169)(126 170)(127 171)(128 172)(129 173)(130 174)(131 175)(132 176)
(1 23)(2 24)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 31)(10 32)(11 33)(12 34)(13 35)(14 36)(15 37)(16 38)(17 39)(18 40)(19 41)(20 42)(21 43)(22 44)(45 88)(46 67)(47 68)(48 69)(49 70)(50 71)(51 72)(52 73)(53 74)(54 75)(55 76)(56 77)(57 78)(58 79)(59 80)(60 81)(61 82)(62 83)(63 84)(64 85)(65 86)(66 87)(89 112)(90 113)(91 114)(92 115)(93 116)(94 117)(95 118)(96 119)(97 120)(98 121)(99 122)(100 123)(101 124)(102 125)(103 126)(104 127)(105 128)(106 129)(107 130)(108 131)(109 132)(110 111)(133 176)(134 155)(135 156)(136 157)(137 158)(138 159)(139 160)(140 161)(141 162)(142 163)(143 164)(144 165)(145 166)(146 167)(147 168)(148 169)(149 170)(150 171)(151 172)(152 173)(153 174)(154 175)
(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)
G:=sub<Sym(176)| (1,108)(2,109)(3,110)(4,89)(5,90)(6,91)(7,92)(8,93)(9,94)(10,95)(11,96)(12,97)(13,98)(14,99)(15,100)(16,101)(17,102)(18,103)(19,104)(20,105)(21,106)(22,107)(23,131)(24,132)(25,111)(26,112)(27,113)(28,114)(29,115)(30,116)(31,117)(32,118)(33,119)(34,120)(35,121)(36,122)(37,123)(38,124)(39,125)(40,126)(41,127)(42,128)(43,129)(44,130)(45,136)(46,137)(47,138)(48,139)(49,140)(50,141)(51,142)(52,143)(53,144)(54,145)(55,146)(56,147)(57,148)(58,149)(59,150)(60,151)(61,152)(62,153)(63,154)(64,133)(65,134)(66,135)(67,158)(68,159)(69,160)(70,161)(71,162)(72,163)(73,164)(74,165)(75,166)(76,167)(77,168)(78,169)(79,170)(80,171)(81,172)(82,173)(83,174)(84,175)(85,176)(86,155)(87,156)(88,157), (1,63)(2,64)(3,65)(4,66)(5,45)(6,46)(7,47)(8,48)(9,49)(10,50)(11,51)(12,52)(13,53)(14,54)(15,55)(16,56)(17,57)(18,58)(19,59)(20,60)(21,61)(22,62)(23,84)(24,85)(25,86)(26,87)(27,88)(28,67)(29,68)(30,69)(31,70)(32,71)(33,72)(34,73)(35,74)(36,75)(37,76)(38,77)(39,78)(40,79)(41,80)(42,81)(43,82)(44,83)(89,135)(90,136)(91,137)(92,138)(93,139)(94,140)(95,141)(96,142)(97,143)(98,144)(99,145)(100,146)(101,147)(102,148)(103,149)(104,150)(105,151)(106,152)(107,153)(108,154)(109,133)(110,134)(111,155)(112,156)(113,157)(114,158)(115,159)(116,160)(117,161)(118,162)(119,163)(120,164)(121,165)(122,166)(123,167)(124,168)(125,169)(126,170)(127,171)(128,172)(129,173)(130,174)(131,175)(132,176), (1,23)(2,24)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,31)(10,32)(11,33)(12,34)(13,35)(14,36)(15,37)(16,38)(17,39)(18,40)(19,41)(20,42)(21,43)(22,44)(45,88)(46,67)(47,68)(48,69)(49,70)(50,71)(51,72)(52,73)(53,74)(54,75)(55,76)(56,77)(57,78)(58,79)(59,80)(60,81)(61,82)(62,83)(63,84)(64,85)(65,86)(66,87)(89,112)(90,113)(91,114)(92,115)(93,116)(94,117)(95,118)(96,119)(97,120)(98,121)(99,122)(100,123)(101,124)(102,125)(103,126)(104,127)(105,128)(106,129)(107,130)(108,131)(109,132)(110,111)(133,176)(134,155)(135,156)(136,157)(137,158)(138,159)(139,160)(140,161)(141,162)(142,163)(143,164)(144,165)(145,166)(146,167)(147,168)(148,169)(149,170)(150,171)(151,172)(152,173)(153,174)(154,175), (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)>;
G:=Group( (1,108)(2,109)(3,110)(4,89)(5,90)(6,91)(7,92)(8,93)(9,94)(10,95)(11,96)(12,97)(13,98)(14,99)(15,100)(16,101)(17,102)(18,103)(19,104)(20,105)(21,106)(22,107)(23,131)(24,132)(25,111)(26,112)(27,113)(28,114)(29,115)(30,116)(31,117)(32,118)(33,119)(34,120)(35,121)(36,122)(37,123)(38,124)(39,125)(40,126)(41,127)(42,128)(43,129)(44,130)(45,136)(46,137)(47,138)(48,139)(49,140)(50,141)(51,142)(52,143)(53,144)(54,145)(55,146)(56,147)(57,148)(58,149)(59,150)(60,151)(61,152)(62,153)(63,154)(64,133)(65,134)(66,135)(67,158)(68,159)(69,160)(70,161)(71,162)(72,163)(73,164)(74,165)(75,166)(76,167)(77,168)(78,169)(79,170)(80,171)(81,172)(82,173)(83,174)(84,175)(85,176)(86,155)(87,156)(88,157), (1,63)(2,64)(3,65)(4,66)(5,45)(6,46)(7,47)(8,48)(9,49)(10,50)(11,51)(12,52)(13,53)(14,54)(15,55)(16,56)(17,57)(18,58)(19,59)(20,60)(21,61)(22,62)(23,84)(24,85)(25,86)(26,87)(27,88)(28,67)(29,68)(30,69)(31,70)(32,71)(33,72)(34,73)(35,74)(36,75)(37,76)(38,77)(39,78)(40,79)(41,80)(42,81)(43,82)(44,83)(89,135)(90,136)(91,137)(92,138)(93,139)(94,140)(95,141)(96,142)(97,143)(98,144)(99,145)(100,146)(101,147)(102,148)(103,149)(104,150)(105,151)(106,152)(107,153)(108,154)(109,133)(110,134)(111,155)(112,156)(113,157)(114,158)(115,159)(116,160)(117,161)(118,162)(119,163)(120,164)(121,165)(122,166)(123,167)(124,168)(125,169)(126,170)(127,171)(128,172)(129,173)(130,174)(131,175)(132,176), (1,23)(2,24)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,31)(10,32)(11,33)(12,34)(13,35)(14,36)(15,37)(16,38)(17,39)(18,40)(19,41)(20,42)(21,43)(22,44)(45,88)(46,67)(47,68)(48,69)(49,70)(50,71)(51,72)(52,73)(53,74)(54,75)(55,76)(56,77)(57,78)(58,79)(59,80)(60,81)(61,82)(62,83)(63,84)(64,85)(65,86)(66,87)(89,112)(90,113)(91,114)(92,115)(93,116)(94,117)(95,118)(96,119)(97,120)(98,121)(99,122)(100,123)(101,124)(102,125)(103,126)(104,127)(105,128)(106,129)(107,130)(108,131)(109,132)(110,111)(133,176)(134,155)(135,156)(136,157)(137,158)(138,159)(139,160)(140,161)(141,162)(142,163)(143,164)(144,165)(145,166)(146,167)(147,168)(148,169)(149,170)(150,171)(151,172)(152,173)(153,174)(154,175), (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) );
G=PermutationGroup([[(1,108),(2,109),(3,110),(4,89),(5,90),(6,91),(7,92),(8,93),(9,94),(10,95),(11,96),(12,97),(13,98),(14,99),(15,100),(16,101),(17,102),(18,103),(19,104),(20,105),(21,106),(22,107),(23,131),(24,132),(25,111),(26,112),(27,113),(28,114),(29,115),(30,116),(31,117),(32,118),(33,119),(34,120),(35,121),(36,122),(37,123),(38,124),(39,125),(40,126),(41,127),(42,128),(43,129),(44,130),(45,136),(46,137),(47,138),(48,139),(49,140),(50,141),(51,142),(52,143),(53,144),(54,145),(55,146),(56,147),(57,148),(58,149),(59,150),(60,151),(61,152),(62,153),(63,154),(64,133),(65,134),(66,135),(67,158),(68,159),(69,160),(70,161),(71,162),(72,163),(73,164),(74,165),(75,166),(76,167),(77,168),(78,169),(79,170),(80,171),(81,172),(82,173),(83,174),(84,175),(85,176),(86,155),(87,156),(88,157)], [(1,63),(2,64),(3,65),(4,66),(5,45),(6,46),(7,47),(8,48),(9,49),(10,50),(11,51),(12,52),(13,53),(14,54),(15,55),(16,56),(17,57),(18,58),(19,59),(20,60),(21,61),(22,62),(23,84),(24,85),(25,86),(26,87),(27,88),(28,67),(29,68),(30,69),(31,70),(32,71),(33,72),(34,73),(35,74),(36,75),(37,76),(38,77),(39,78),(40,79),(41,80),(42,81),(43,82),(44,83),(89,135),(90,136),(91,137),(92,138),(93,139),(94,140),(95,141),(96,142),(97,143),(98,144),(99,145),(100,146),(101,147),(102,148),(103,149),(104,150),(105,151),(106,152),(107,153),(108,154),(109,133),(110,134),(111,155),(112,156),(113,157),(114,158),(115,159),(116,160),(117,161),(118,162),(119,163),(120,164),(121,165),(122,166),(123,167),(124,168),(125,169),(126,170),(127,171),(128,172),(129,173),(130,174),(131,175),(132,176)], [(1,23),(2,24),(3,25),(4,26),(5,27),(6,28),(7,29),(8,30),(9,31),(10,32),(11,33),(12,34),(13,35),(14,36),(15,37),(16,38),(17,39),(18,40),(19,41),(20,42),(21,43),(22,44),(45,88),(46,67),(47,68),(48,69),(49,70),(50,71),(51,72),(52,73),(53,74),(54,75),(55,76),(56,77),(57,78),(58,79),(59,80),(60,81),(61,82),(62,83),(63,84),(64,85),(65,86),(66,87),(89,112),(90,113),(91,114),(92,115),(93,116),(94,117),(95,118),(96,119),(97,120),(98,121),(99,122),(100,123),(101,124),(102,125),(103,126),(104,127),(105,128),(106,129),(107,130),(108,131),(109,132),(110,111),(133,176),(134,155),(135,156),(136,157),(137,158),(138,159),(139,160),(140,161),(141,162),(142,163),(143,164),(144,165),(145,166),(146,167),(147,168),(148,169),(149,170),(150,171),(151,172),(152,173),(153,174),(154,175)], [(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)]])
C23×C22 is a maximal subgroup of
C24⋊D11
176 conjugacy classes
| class | 1 | 2A | ··· | 2O | 11A | ··· | 11J | 22A | ··· | 22ET |
| order | 1 | 2 | ··· | 2 | 11 | ··· | 11 | 22 | ··· | 22 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
176 irreducible representations
| dim | 1 | 1 | 1 | 1 |
| type | + | + | ||
| image | C1 | C2 | C11 | C22 |
| kernel | C23×C22 | C22×C22 | C24 | C23 |
| # reps | 1 | 15 | 10 | 150 |
Matrix representation of C23×C22 ►in GL4(𝔽23) generated by
| 1 | 0 | 0 | 0 |
| 0 | 22 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 22 |
| 22 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 22 |
| 1 | 0 | 0 | 0 |
| 0 | 22 | 0 | 0 |
| 0 | 0 | 22 | 0 |
| 0 | 0 | 0 | 1 |
| 3 | 0 | 0 | 0 |
| 0 | 11 | 0 | 0 |
| 0 | 0 | 15 | 0 |
| 0 | 0 | 0 | 21 |
G:=sub<GL(4,GF(23))| [1,0,0,0,0,22,0,0,0,0,1,0,0,0,0,22],[22,0,0,0,0,1,0,0,0,0,1,0,0,0,0,22],[1,0,0,0,0,22,0,0,0,0,22,0,0,0,0,1],[3,0,0,0,0,11,0,0,0,0,15,0,0,0,0,21] >;
C23×C22 in GAP, Magma, Sage, TeX
C_2^3\times C_{22} % in TeX
G:=Group("C2^3xC22"); // GroupNames label
G:=SmallGroup(176,42);
// by ID
G=gap.SmallGroup(176,42);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-11]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^2=d^22=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