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G = C2×C90order 180 = 22·32·5

Abelian group of type [2,90]

direct product, abelian, monomial, 2-elementary

Aliases: C2×C90, SmallGroup(180,12)

Series: Derived Chief Lower central Upper central

C1 — C2×C90
C1C3C15C45C90 — C2×C90
C1 — C2×C90
C1 — C2×C90

Generators and relations for C2×C90
 G = < a,b | a2=b90=1, ab=ba >


Smallest permutation representation of C2×C90
Regular action on 180 points
Generators in S180
(1 142)(2 143)(3 144)(4 145)(5 146)(6 147)(7 148)(8 149)(9 150)(10 151)(11 152)(12 153)(13 154)(14 155)(15 156)(16 157)(17 158)(18 159)(19 160)(20 161)(21 162)(22 163)(23 164)(24 165)(25 166)(26 167)(27 168)(28 169)(29 170)(30 171)(31 172)(32 173)(33 174)(34 175)(35 176)(36 177)(37 178)(38 179)(39 180)(40 91)(41 92)(42 93)(43 94)(44 95)(45 96)(46 97)(47 98)(48 99)(49 100)(50 101)(51 102)(52 103)(53 104)(54 105)(55 106)(56 107)(57 108)(58 109)(59 110)(60 111)(61 112)(62 113)(63 114)(64 115)(65 116)(66 117)(67 118)(68 119)(69 120)(70 121)(71 122)(72 123)(73 124)(74 125)(75 126)(76 127)(77 128)(78 129)(79 130)(80 131)(81 132)(82 133)(83 134)(84 135)(85 136)(86 137)(87 138)(88 139)(89 140)(90 141)
(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)

G:=sub<Sym(180)| (1,142)(2,143)(3,144)(4,145)(5,146)(6,147)(7,148)(8,149)(9,150)(10,151)(11,152)(12,153)(13,154)(14,155)(15,156)(16,157)(17,158)(18,159)(19,160)(20,161)(21,162)(22,163)(23,164)(24,165)(25,166)(26,167)(27,168)(28,169)(29,170)(30,171)(31,172)(32,173)(33,174)(34,175)(35,176)(36,177)(37,178)(38,179)(39,180)(40,91)(41,92)(42,93)(43,94)(44,95)(45,96)(46,97)(47,98)(48,99)(49,100)(50,101)(51,102)(52,103)(53,104)(54,105)(55,106)(56,107)(57,108)(58,109)(59,110)(60,111)(61,112)(62,113)(63,114)(64,115)(65,116)(66,117)(67,118)(68,119)(69,120)(70,121)(71,122)(72,123)(73,124)(74,125)(75,126)(76,127)(77,128)(78,129)(79,130)(80,131)(81,132)(82,133)(83,134)(84,135)(85,136)(86,137)(87,138)(88,139)(89,140)(90,141), (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)>;

G:=Group( (1,142)(2,143)(3,144)(4,145)(5,146)(6,147)(7,148)(8,149)(9,150)(10,151)(11,152)(12,153)(13,154)(14,155)(15,156)(16,157)(17,158)(18,159)(19,160)(20,161)(21,162)(22,163)(23,164)(24,165)(25,166)(26,167)(27,168)(28,169)(29,170)(30,171)(31,172)(32,173)(33,174)(34,175)(35,176)(36,177)(37,178)(38,179)(39,180)(40,91)(41,92)(42,93)(43,94)(44,95)(45,96)(46,97)(47,98)(48,99)(49,100)(50,101)(51,102)(52,103)(53,104)(54,105)(55,106)(56,107)(57,108)(58,109)(59,110)(60,111)(61,112)(62,113)(63,114)(64,115)(65,116)(66,117)(67,118)(68,119)(69,120)(70,121)(71,122)(72,123)(73,124)(74,125)(75,126)(76,127)(77,128)(78,129)(79,130)(80,131)(81,132)(82,133)(83,134)(84,135)(85,136)(86,137)(87,138)(88,139)(89,140)(90,141), (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) );

G=PermutationGroup([(1,142),(2,143),(3,144),(4,145),(5,146),(6,147),(7,148),(8,149),(9,150),(10,151),(11,152),(12,153),(13,154),(14,155),(15,156),(16,157),(17,158),(18,159),(19,160),(20,161),(21,162),(22,163),(23,164),(24,165),(25,166),(26,167),(27,168),(28,169),(29,170),(30,171),(31,172),(32,173),(33,174),(34,175),(35,176),(36,177),(37,178),(38,179),(39,180),(40,91),(41,92),(42,93),(43,94),(44,95),(45,96),(46,97),(47,98),(48,99),(49,100),(50,101),(51,102),(52,103),(53,104),(54,105),(55,106),(56,107),(57,108),(58,109),(59,110),(60,111),(61,112),(62,113),(63,114),(64,115),(65,116),(66,117),(67,118),(68,119),(69,120),(70,121),(71,122),(72,123),(73,124),(74,125),(75,126),(76,127),(77,128),(78,129),(79,130),(80,131),(81,132),(82,133),(83,134),(84,135),(85,136),(86,137),(87,138),(88,139),(89,140),(90,141)], [(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)])

C2×C90 is a maximal subgroup of   C457D4

180 conjugacy classes

class 1 2A2B2C3A3B5A5B5C5D6A···6F9A···9F10A···10L15A···15H18A···18R30A···30X45A···45X90A···90BT
order12223355556···69···910···1015···1518···1830···3045···4590···90
size11111111111···11···11···11···11···11···11···11···1

180 irreducible representations

dim111111111111
type++
imageC1C2C3C5C6C9C10C15C18C30C45C90
kernelC2×C90C90C2×C30C2×C18C30C2×C10C18C2×C6C10C6C22C2
# reps13246612818242472

Matrix representation of C2×C90 in GL2(𝔽181) generated by

1800
0180
,
90
0147
G:=sub<GL(2,GF(181))| [180,0,0,180],[9,0,0,147] >;

C2×C90 in GAP, Magma, Sage, TeX

C_2\times C_{90}
% in TeX

G:=Group("C2xC90");
// GroupNames label

G:=SmallGroup(180,12);
// by ID

G=gap.SmallGroup(180,12);
# by ID

G:=PCGroup([5,-2,-2,-3,-5,-3,237]);
// Polycyclic

G:=Group<a,b|a^2=b^90=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C2×C90 in TeX

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