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## G = C22×C44order 176 = 24·11

### Abelian group of type [2,2,44]

Aliases: C22×C44, SmallGroup(176,37)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C44
 Chief series C1 — C2 — C22 — C44 — C2×C44 — C22×C44
 Lower central C1 — C22×C44
 Upper central 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 ··· 2G 4A ··· 4H 11A ··· 11J 22A ··· 22BR 44A ··· 44CB order 1 2 ··· 2 4 ··· 4 11 ··· 11 22 ··· 22 44 ··· 44 size 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1

176 irreducible representations

 dim 1 1 1 1 1 1 1 1 type + + + image C1 C2 C2 C4 C11 C22 C22 C44 kernel C22×C44 C2×C44 C22×C22 C2×C22 C22×C4 C2×C4 C23 C22 # reps 1 6 1 8 10 60 10 80

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

 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
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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