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G = C22×C44order 176 = 24·11

Abelian group of type [2,2,44]

direct product, abelian, monomial, 2-elementary

Aliases: C22×C44, SmallGroup(176,37)

Series: Derived Chief Lower central Upper central

C1 — C22×C44
C1C2C22C44C2×C44 — C22×C44
C1 — C22×C44
C1 — C22×C44

Generators and relations for C22×C44
 G = < a,b,c | a2=b2=c44=1, ab=ba, ac=ca, bc=cb >

Subgroups: 54, all normal (8 characteristic)
C1, C2, C2 [×6], C4 [×4], C22 [×7], C2×C4 [×6], C23, C11, C22×C4, C22, C22 [×6], C44 [×4], C2×C22 [×7], C2×C44 [×6], C22×C22, C22×C44
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], C23, C11, C22×C4, C22 [×7], C44 [×4], C2×C22 [×7], C2×C44 [×6], C22×C22, C22×C44

Smallest permutation representation of C22×C44
Regular action on 176 points
Generators in S176
(1 119)(2 120)(3 121)(4 122)(5 123)(6 124)(7 125)(8 126)(9 127)(10 128)(11 129)(12 130)(13 131)(14 132)(15 89)(16 90)(17 91)(18 92)(19 93)(20 94)(21 95)(22 96)(23 97)(24 98)(25 99)(26 100)(27 101)(28 102)(29 103)(30 104)(31 105)(32 106)(33 107)(34 108)(35 109)(36 110)(37 111)(38 112)(39 113)(40 114)(41 115)(42 116)(43 117)(44 118)(45 163)(46 164)(47 165)(48 166)(49 167)(50 168)(51 169)(52 170)(53 171)(54 172)(55 173)(56 174)(57 175)(58 176)(59 133)(60 134)(61 135)(62 136)(63 137)(64 138)(65 139)(66 140)(67 141)(68 142)(69 143)(70 144)(71 145)(72 146)(73 147)(74 148)(75 149)(76 150)(77 151)(78 152)(79 153)(80 154)(81 155)(82 156)(83 157)(84 158)(85 159)(86 160)(87 161)(88 162)
(1 143)(2 144)(3 145)(4 146)(5 147)(6 148)(7 149)(8 150)(9 151)(10 152)(11 153)(12 154)(13 155)(14 156)(15 157)(16 158)(17 159)(18 160)(19 161)(20 162)(21 163)(22 164)(23 165)(24 166)(25 167)(26 168)(27 169)(28 170)(29 171)(30 172)(31 173)(32 174)(33 175)(34 176)(35 133)(36 134)(37 135)(38 136)(39 137)(40 138)(41 139)(42 140)(43 141)(44 142)(45 95)(46 96)(47 97)(48 98)(49 99)(50 100)(51 101)(52 102)(53 103)(54 104)(55 105)(56 106)(57 107)(58 108)(59 109)(60 110)(61 111)(62 112)(63 113)(64 114)(65 115)(66 116)(67 117)(68 118)(69 119)(70 120)(71 121)(72 122)(73 123)(74 124)(75 125)(76 126)(77 127)(78 128)(79 129)(80 130)(81 131)(82 132)(83 89)(84 90)(85 91)(86 92)(87 93)(88 94)
(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)| (1,119)(2,120)(3,121)(4,122)(5,123)(6,124)(7,125)(8,126)(9,127)(10,128)(11,129)(12,130)(13,131)(14,132)(15,89)(16,90)(17,91)(18,92)(19,93)(20,94)(21,95)(22,96)(23,97)(24,98)(25,99)(26,100)(27,101)(28,102)(29,103)(30,104)(31,105)(32,106)(33,107)(34,108)(35,109)(36,110)(37,111)(38,112)(39,113)(40,114)(41,115)(42,116)(43,117)(44,118)(45,163)(46,164)(47,165)(48,166)(49,167)(50,168)(51,169)(52,170)(53,171)(54,172)(55,173)(56,174)(57,175)(58,176)(59,133)(60,134)(61,135)(62,136)(63,137)(64,138)(65,139)(66,140)(67,141)(68,142)(69,143)(70,144)(71,145)(72,146)(73,147)(74,148)(75,149)(76,150)(77,151)(78,152)(79,153)(80,154)(81,155)(82,156)(83,157)(84,158)(85,159)(86,160)(87,161)(88,162), (1,143)(2,144)(3,145)(4,146)(5,147)(6,148)(7,149)(8,150)(9,151)(10,152)(11,153)(12,154)(13,155)(14,156)(15,157)(16,158)(17,159)(18,160)(19,161)(20,162)(21,163)(22,164)(23,165)(24,166)(25,167)(26,168)(27,169)(28,170)(29,171)(30,172)(31,173)(32,174)(33,175)(34,176)(35,133)(36,134)(37,135)(38,136)(39,137)(40,138)(41,139)(42,140)(43,141)(44,142)(45,95)(46,96)(47,97)(48,98)(49,99)(50,100)(51,101)(52,102)(53,103)(54,104)(55,105)(56,106)(57,107)(58,108)(59,109)(60,110)(61,111)(62,112)(63,113)(64,114)(65,115)(66,116)(67,117)(68,118)(69,119)(70,120)(71,121)(72,122)(73,123)(74,124)(75,125)(76,126)(77,127)(78,128)(79,129)(80,130)(81,131)(82,132)(83,89)(84,90)(85,91)(86,92)(87,93)(88,94), (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( (1,119)(2,120)(3,121)(4,122)(5,123)(6,124)(7,125)(8,126)(9,127)(10,128)(11,129)(12,130)(13,131)(14,132)(15,89)(16,90)(17,91)(18,92)(19,93)(20,94)(21,95)(22,96)(23,97)(24,98)(25,99)(26,100)(27,101)(28,102)(29,103)(30,104)(31,105)(32,106)(33,107)(34,108)(35,109)(36,110)(37,111)(38,112)(39,113)(40,114)(41,115)(42,116)(43,117)(44,118)(45,163)(46,164)(47,165)(48,166)(49,167)(50,168)(51,169)(52,170)(53,171)(54,172)(55,173)(56,174)(57,175)(58,176)(59,133)(60,134)(61,135)(62,136)(63,137)(64,138)(65,139)(66,140)(67,141)(68,142)(69,143)(70,144)(71,145)(72,146)(73,147)(74,148)(75,149)(76,150)(77,151)(78,152)(79,153)(80,154)(81,155)(82,156)(83,157)(84,158)(85,159)(86,160)(87,161)(88,162), (1,143)(2,144)(3,145)(4,146)(5,147)(6,148)(7,149)(8,150)(9,151)(10,152)(11,153)(12,154)(13,155)(14,156)(15,157)(16,158)(17,159)(18,160)(19,161)(20,162)(21,163)(22,164)(23,165)(24,166)(25,167)(26,168)(27,169)(28,170)(29,171)(30,172)(31,173)(32,174)(33,175)(34,176)(35,133)(36,134)(37,135)(38,136)(39,137)(40,138)(41,139)(42,140)(43,141)(44,142)(45,95)(46,96)(47,97)(48,98)(49,99)(50,100)(51,101)(52,102)(53,103)(54,104)(55,105)(56,106)(57,107)(58,108)(59,109)(60,110)(61,111)(62,112)(63,113)(64,114)(65,115)(66,116)(67,117)(68,118)(69,119)(70,120)(71,121)(72,122)(73,123)(74,124)(75,125)(76,126)(77,127)(78,128)(79,129)(80,130)(81,131)(82,132)(83,89)(84,90)(85,91)(86,92)(87,93)(88,94), (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([(1,119),(2,120),(3,121),(4,122),(5,123),(6,124),(7,125),(8,126),(9,127),(10,128),(11,129),(12,130),(13,131),(14,132),(15,89),(16,90),(17,91),(18,92),(19,93),(20,94),(21,95),(22,96),(23,97),(24,98),(25,99),(26,100),(27,101),(28,102),(29,103),(30,104),(31,105),(32,106),(33,107),(34,108),(35,109),(36,110),(37,111),(38,112),(39,113),(40,114),(41,115),(42,116),(43,117),(44,118),(45,163),(46,164),(47,165),(48,166),(49,167),(50,168),(51,169),(52,170),(53,171),(54,172),(55,173),(56,174),(57,175),(58,176),(59,133),(60,134),(61,135),(62,136),(63,137),(64,138),(65,139),(66,140),(67,141),(68,142),(69,143),(70,144),(71,145),(72,146),(73,147),(74,148),(75,149),(76,150),(77,151),(78,152),(79,153),(80,154),(81,155),(82,156),(83,157),(84,158),(85,159),(86,160),(87,161),(88,162)], [(1,143),(2,144),(3,145),(4,146),(5,147),(6,148),(7,149),(8,150),(9,151),(10,152),(11,153),(12,154),(13,155),(14,156),(15,157),(16,158),(17,159),(18,160),(19,161),(20,162),(21,163),(22,164),(23,165),(24,166),(25,167),(26,168),(27,169),(28,170),(29,171),(30,172),(31,173),(32,174),(33,175),(34,176),(35,133),(36,134),(37,135),(38,136),(39,137),(40,138),(41,139),(42,140),(43,141),(44,142),(45,95),(46,96),(47,97),(48,98),(49,99),(50,100),(51,101),(52,102),(53,103),(54,104),(55,105),(56,106),(57,107),(58,108),(59,109),(60,110),(61,111),(62,112),(63,113),(64,114),(65,115),(66,116),(67,117),(68,118),(69,119),(70,120),(71,121),(72,122),(73,123),(74,124),(75,125),(76,126),(77,127),(78,128),(79,129),(80,130),(81,131),(82,132),(83,89),(84,90),(85,91),(86,92),(87,93),(88,94)], [(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)])

C22×C44 is a maximal subgroup of   C44.55D4  C22.C42  C44.48D4  C23.21D22  C23.23D22  C447D4

176 conjugacy classes

class 1 2A···2G4A···4H11A···11J22A···22BR44A···44CB
order12···24···411···1122···2244···44
size11···11···11···11···11···1

176 irreducible representations

dim11111111
type+++
imageC1C2C2C4C11C22C22C44
kernelC22×C44C2×C44C22×C22C2×C22C22×C4C2×C4C23C22
# reps161810601080

Matrix representation of C22×C44 in GL3(𝔽89) generated by

8800
010
0088
,
100
0880
0088
,
5300
090
0072
G:=sub<GL(3,GF(89))| [88,0,0,0,1,0,0,0,88],[1,0,0,0,88,0,0,0,88],[53,0,0,0,9,0,0,0,72] >;

C22×C44 in GAP, Magma, Sage, TeX

C_2^2\times C_{44}
% in TeX

G:=Group("C2^2xC44");
// GroupNames label

G:=SmallGroup(176,37);
// by ID

G=gap.SmallGroup(176,37);
# by ID

G:=PCGroup([5,-2,-2,-2,-11,-2,440]);
// Polycyclic

G:=Group<a,b,c|a^2=b^2=c^44=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations

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