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G = C23×C22order 176 = 24·11

Abelian group of type [2,2,2,22]

direct product, abelian, monomial, 2-elementary

Aliases: C23×C22, SmallGroup(176,42)

Series: Derived Chief Lower central Upper central

C1 — C23×C22
C1C11C22C2×C22C22×C22 — 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 [×15], C22 [×35], C23 [×15], C11, C24, C22 [×15], C2×C22 [×35], C22×C22 [×15], C23×C22
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], C11, C24, C22 [×15], C2×C22 [×35], C22×C22 [×15], C23×C22

Smallest permutation representation of C23×C22
Regular action on 176 points
Generators in S176
(1 104)(2 105)(3 106)(4 107)(5 108)(6 109)(7 110)(8 89)(9 90)(10 91)(11 92)(12 93)(13 94)(14 95)(15 96)(16 97)(17 98)(18 99)(19 100)(20 101)(21 102)(22 103)(23 111)(24 112)(25 113)(26 114)(27 115)(28 116)(29 117)(30 118)(31 119)(32 120)(33 121)(34 122)(35 123)(36 124)(37 125)(38 126)(39 127)(40 128)(41 129)(42 130)(43 131)(44 132)(45 147)(46 148)(47 149)(48 150)(49 151)(50 152)(51 153)(52 154)(53 133)(54 134)(55 135)(56 136)(57 137)(58 138)(59 139)(60 140)(61 141)(62 142)(63 143)(64 144)(65 145)(66 146)(67 161)(68 162)(69 163)(70 164)(71 165)(72 166)(73 167)(74 168)(75 169)(76 170)(77 171)(78 172)(79 173)(80 174)(81 175)(82 176)(83 155)(84 156)(85 157)(86 158)(87 159)(88 160)
(1 65)(2 66)(3 45)(4 46)(5 47)(6 48)(7 49)(8 50)(9 51)(10 52)(11 53)(12 54)(13 55)(14 56)(15 57)(16 58)(17 59)(18 60)(19 61)(20 62)(21 63)(22 64)(23 69)(24 70)(25 71)(26 72)(27 73)(28 74)(29 75)(30 76)(31 77)(32 78)(33 79)(34 80)(35 81)(36 82)(37 83)(38 84)(39 85)(40 86)(41 87)(42 88)(43 67)(44 68)(89 152)(90 153)(91 154)(92 133)(93 134)(94 135)(95 136)(96 137)(97 138)(98 139)(99 140)(100 141)(101 142)(102 143)(103 144)(104 145)(105 146)(106 147)(107 148)(108 149)(109 150)(110 151)(111 163)(112 164)(113 165)(114 166)(115 167)(116 168)(117 169)(118 170)(119 171)(120 172)(121 173)(122 174)(123 175)(124 176)(125 155)(126 156)(127 157)(128 158)(129 159)(130 160)(131 161)(132 162)
(1 35)(2 36)(3 37)(4 38)(5 39)(6 40)(7 41)(8 42)(9 43)(10 44)(11 23)(12 24)(13 25)(14 26)(15 27)(16 28)(17 29)(18 30)(19 31)(20 32)(21 33)(22 34)(45 83)(46 84)(47 85)(48 86)(49 87)(50 88)(51 67)(52 68)(53 69)(54 70)(55 71)(56 72)(57 73)(58 74)(59 75)(60 76)(61 77)(62 78)(63 79)(64 80)(65 81)(66 82)(89 130)(90 131)(91 132)(92 111)(93 112)(94 113)(95 114)(96 115)(97 116)(98 117)(99 118)(100 119)(101 120)(102 121)(103 122)(104 123)(105 124)(106 125)(107 126)(108 127)(109 128)(110 129)(133 163)(134 164)(135 165)(136 166)(137 167)(138 168)(139 169)(140 170)(141 171)(142 172)(143 173)(144 174)(145 175)(146 176)(147 155)(148 156)(149 157)(150 158)(151 159)(152 160)(153 161)(154 162)
(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)| (1,104)(2,105)(3,106)(4,107)(5,108)(6,109)(7,110)(8,89)(9,90)(10,91)(11,92)(12,93)(13,94)(14,95)(15,96)(16,97)(17,98)(18,99)(19,100)(20,101)(21,102)(22,103)(23,111)(24,112)(25,113)(26,114)(27,115)(28,116)(29,117)(30,118)(31,119)(32,120)(33,121)(34,122)(35,123)(36,124)(37,125)(38,126)(39,127)(40,128)(41,129)(42,130)(43,131)(44,132)(45,147)(46,148)(47,149)(48,150)(49,151)(50,152)(51,153)(52,154)(53,133)(54,134)(55,135)(56,136)(57,137)(58,138)(59,139)(60,140)(61,141)(62,142)(63,143)(64,144)(65,145)(66,146)(67,161)(68,162)(69,163)(70,164)(71,165)(72,166)(73,167)(74,168)(75,169)(76,170)(77,171)(78,172)(79,173)(80,174)(81,175)(82,176)(83,155)(84,156)(85,157)(86,158)(87,159)(88,160), (1,65)(2,66)(3,45)(4,46)(5,47)(6,48)(7,49)(8,50)(9,51)(10,52)(11,53)(12,54)(13,55)(14,56)(15,57)(16,58)(17,59)(18,60)(19,61)(20,62)(21,63)(22,64)(23,69)(24,70)(25,71)(26,72)(27,73)(28,74)(29,75)(30,76)(31,77)(32,78)(33,79)(34,80)(35,81)(36,82)(37,83)(38,84)(39,85)(40,86)(41,87)(42,88)(43,67)(44,68)(89,152)(90,153)(91,154)(92,133)(93,134)(94,135)(95,136)(96,137)(97,138)(98,139)(99,140)(100,141)(101,142)(102,143)(103,144)(104,145)(105,146)(106,147)(107,148)(108,149)(109,150)(110,151)(111,163)(112,164)(113,165)(114,166)(115,167)(116,168)(117,169)(118,170)(119,171)(120,172)(121,173)(122,174)(123,175)(124,176)(125,155)(126,156)(127,157)(128,158)(129,159)(130,160)(131,161)(132,162), (1,35)(2,36)(3,37)(4,38)(5,39)(6,40)(7,41)(8,42)(9,43)(10,44)(11,23)(12,24)(13,25)(14,26)(15,27)(16,28)(17,29)(18,30)(19,31)(20,32)(21,33)(22,34)(45,83)(46,84)(47,85)(48,86)(49,87)(50,88)(51,67)(52,68)(53,69)(54,70)(55,71)(56,72)(57,73)(58,74)(59,75)(60,76)(61,77)(62,78)(63,79)(64,80)(65,81)(66,82)(89,130)(90,131)(91,132)(92,111)(93,112)(94,113)(95,114)(96,115)(97,116)(98,117)(99,118)(100,119)(101,120)(102,121)(103,122)(104,123)(105,124)(106,125)(107,126)(108,127)(109,128)(110,129)(133,163)(134,164)(135,165)(136,166)(137,167)(138,168)(139,169)(140,170)(141,171)(142,172)(143,173)(144,174)(145,175)(146,176)(147,155)(148,156)(149,157)(150,158)(151,159)(152,160)(153,161)(154,162), (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( (1,104)(2,105)(3,106)(4,107)(5,108)(6,109)(7,110)(8,89)(9,90)(10,91)(11,92)(12,93)(13,94)(14,95)(15,96)(16,97)(17,98)(18,99)(19,100)(20,101)(21,102)(22,103)(23,111)(24,112)(25,113)(26,114)(27,115)(28,116)(29,117)(30,118)(31,119)(32,120)(33,121)(34,122)(35,123)(36,124)(37,125)(38,126)(39,127)(40,128)(41,129)(42,130)(43,131)(44,132)(45,147)(46,148)(47,149)(48,150)(49,151)(50,152)(51,153)(52,154)(53,133)(54,134)(55,135)(56,136)(57,137)(58,138)(59,139)(60,140)(61,141)(62,142)(63,143)(64,144)(65,145)(66,146)(67,161)(68,162)(69,163)(70,164)(71,165)(72,166)(73,167)(74,168)(75,169)(76,170)(77,171)(78,172)(79,173)(80,174)(81,175)(82,176)(83,155)(84,156)(85,157)(86,158)(87,159)(88,160), (1,65)(2,66)(3,45)(4,46)(5,47)(6,48)(7,49)(8,50)(9,51)(10,52)(11,53)(12,54)(13,55)(14,56)(15,57)(16,58)(17,59)(18,60)(19,61)(20,62)(21,63)(22,64)(23,69)(24,70)(25,71)(26,72)(27,73)(28,74)(29,75)(30,76)(31,77)(32,78)(33,79)(34,80)(35,81)(36,82)(37,83)(38,84)(39,85)(40,86)(41,87)(42,88)(43,67)(44,68)(89,152)(90,153)(91,154)(92,133)(93,134)(94,135)(95,136)(96,137)(97,138)(98,139)(99,140)(100,141)(101,142)(102,143)(103,144)(104,145)(105,146)(106,147)(107,148)(108,149)(109,150)(110,151)(111,163)(112,164)(113,165)(114,166)(115,167)(116,168)(117,169)(118,170)(119,171)(120,172)(121,173)(122,174)(123,175)(124,176)(125,155)(126,156)(127,157)(128,158)(129,159)(130,160)(131,161)(132,162), (1,35)(2,36)(3,37)(4,38)(5,39)(6,40)(7,41)(8,42)(9,43)(10,44)(11,23)(12,24)(13,25)(14,26)(15,27)(16,28)(17,29)(18,30)(19,31)(20,32)(21,33)(22,34)(45,83)(46,84)(47,85)(48,86)(49,87)(50,88)(51,67)(52,68)(53,69)(54,70)(55,71)(56,72)(57,73)(58,74)(59,75)(60,76)(61,77)(62,78)(63,79)(64,80)(65,81)(66,82)(89,130)(90,131)(91,132)(92,111)(93,112)(94,113)(95,114)(96,115)(97,116)(98,117)(99,118)(100,119)(101,120)(102,121)(103,122)(104,123)(105,124)(106,125)(107,126)(108,127)(109,128)(110,129)(133,163)(134,164)(135,165)(136,166)(137,167)(138,168)(139,169)(140,170)(141,171)(142,172)(143,173)(144,174)(145,175)(146,176)(147,155)(148,156)(149,157)(150,158)(151,159)(152,160)(153,161)(154,162), (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([(1,104),(2,105),(3,106),(4,107),(5,108),(6,109),(7,110),(8,89),(9,90),(10,91),(11,92),(12,93),(13,94),(14,95),(15,96),(16,97),(17,98),(18,99),(19,100),(20,101),(21,102),(22,103),(23,111),(24,112),(25,113),(26,114),(27,115),(28,116),(29,117),(30,118),(31,119),(32,120),(33,121),(34,122),(35,123),(36,124),(37,125),(38,126),(39,127),(40,128),(41,129),(42,130),(43,131),(44,132),(45,147),(46,148),(47,149),(48,150),(49,151),(50,152),(51,153),(52,154),(53,133),(54,134),(55,135),(56,136),(57,137),(58,138),(59,139),(60,140),(61,141),(62,142),(63,143),(64,144),(65,145),(66,146),(67,161),(68,162),(69,163),(70,164),(71,165),(72,166),(73,167),(74,168),(75,169),(76,170),(77,171),(78,172),(79,173),(80,174),(81,175),(82,176),(83,155),(84,156),(85,157),(86,158),(87,159),(88,160)], [(1,65),(2,66),(3,45),(4,46),(5,47),(6,48),(7,49),(8,50),(9,51),(10,52),(11,53),(12,54),(13,55),(14,56),(15,57),(16,58),(17,59),(18,60),(19,61),(20,62),(21,63),(22,64),(23,69),(24,70),(25,71),(26,72),(27,73),(28,74),(29,75),(30,76),(31,77),(32,78),(33,79),(34,80),(35,81),(36,82),(37,83),(38,84),(39,85),(40,86),(41,87),(42,88),(43,67),(44,68),(89,152),(90,153),(91,154),(92,133),(93,134),(94,135),(95,136),(96,137),(97,138),(98,139),(99,140),(100,141),(101,142),(102,143),(103,144),(104,145),(105,146),(106,147),(107,148),(108,149),(109,150),(110,151),(111,163),(112,164),(113,165),(114,166),(115,167),(116,168),(117,169),(118,170),(119,171),(120,172),(121,173),(122,174),(123,175),(124,176),(125,155),(126,156),(127,157),(128,158),(129,159),(130,160),(131,161),(132,162)], [(1,35),(2,36),(3,37),(4,38),(5,39),(6,40),(7,41),(8,42),(9,43),(10,44),(11,23),(12,24),(13,25),(14,26),(15,27),(16,28),(17,29),(18,30),(19,31),(20,32),(21,33),(22,34),(45,83),(46,84),(47,85),(48,86),(49,87),(50,88),(51,67),(52,68),(53,69),(54,70),(55,71),(56,72),(57,73),(58,74),(59,75),(60,76),(61,77),(62,78),(63,79),(64,80),(65,81),(66,82),(89,130),(90,131),(91,132),(92,111),(93,112),(94,113),(95,114),(96,115),(97,116),(98,117),(99,118),(100,119),(101,120),(102,121),(103,122),(104,123),(105,124),(106,125),(107,126),(108,127),(109,128),(110,129),(133,163),(134,164),(135,165),(136,166),(137,167),(138,168),(139,169),(140,170),(141,171),(142,172),(143,173),(144,174),(145,175),(146,176),(147,155),(148,156),(149,157),(150,158),(151,159),(152,160),(153,161),(154,162)], [(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···2O11A···11J22A···22ET
order12···211···1122···22
size11···11···11···1

176 irreducible representations

dim1111
type++
imageC1C2C11C22
kernelC23×C22C22×C22C24C23
# reps11510150

Matrix representation of C23×C22 in GL4(𝔽23) generated by

1000
02200
0010
00022
,
22000
0100
0010
00022
,
1000
02200
00220
0001
,
3000
01100
00150
00021
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

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