Copied to
clipboard

G = C2×C88order 176 = 24·11

Abelian group of type [2,88]

direct product, abelian, monomial, 2-elementary

Aliases: C2×C88, SmallGroup(176,22)

Series: Derived Chief Lower central Upper central

C1 — C2×C88
C1C2C4C44C88 — C2×C88
C1 — C2×C88
C1 — C2×C88

Generators and relations for C2×C88
 G = < a,b | a2=b88=1, ab=ba >


Smallest permutation representation of C2×C88
Regular action on 176 points
Generators in S176
(1 130)(2 131)(3 132)(4 133)(5 134)(6 135)(7 136)(8 137)(9 138)(10 139)(11 140)(12 141)(13 142)(14 143)(15 144)(16 145)(17 146)(18 147)(19 148)(20 149)(21 150)(22 151)(23 152)(24 153)(25 154)(26 155)(27 156)(28 157)(29 158)(30 159)(31 160)(32 161)(33 162)(34 163)(35 164)(36 165)(37 166)(38 167)(39 168)(40 169)(41 170)(42 171)(43 172)(44 173)(45 174)(46 175)(47 176)(48 89)(49 90)(50 91)(51 92)(52 93)(53 94)(54 95)(55 96)(56 97)(57 98)(58 99)(59 100)(60 101)(61 102)(62 103)(63 104)(64 105)(65 106)(66 107)(67 108)(68 109)(69 110)(70 111)(71 112)(72 113)(73 114)(74 115)(75 116)(76 117)(77 118)(78 119)(79 120)(80 121)(81 122)(82 123)(83 124)(84 125)(85 126)(86 127)(87 128)(88 129)
(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)| (1,130)(2,131)(3,132)(4,133)(5,134)(6,135)(7,136)(8,137)(9,138)(10,139)(11,140)(12,141)(13,142)(14,143)(15,144)(16,145)(17,146)(18,147)(19,148)(20,149)(21,150)(22,151)(23,152)(24,153)(25,154)(26,155)(27,156)(28,157)(29,158)(30,159)(31,160)(32,161)(33,162)(34,163)(35,164)(36,165)(37,166)(38,167)(39,168)(40,169)(41,170)(42,171)(43,172)(44,173)(45,174)(46,175)(47,176)(48,89)(49,90)(50,91)(51,92)(52,93)(53,94)(54,95)(55,96)(56,97)(57,98)(58,99)(59,100)(60,101)(61,102)(62,103)(63,104)(64,105)(65,106)(66,107)(67,108)(68,109)(69,110)(70,111)(71,112)(72,113)(73,114)(74,115)(75,116)(76,117)(77,118)(78,119)(79,120)(80,121)(81,122)(82,123)(83,124)(84,125)(85,126)(86,127)(87,128)(88,129), (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( (1,130)(2,131)(3,132)(4,133)(5,134)(6,135)(7,136)(8,137)(9,138)(10,139)(11,140)(12,141)(13,142)(14,143)(15,144)(16,145)(17,146)(18,147)(19,148)(20,149)(21,150)(22,151)(23,152)(24,153)(25,154)(26,155)(27,156)(28,157)(29,158)(30,159)(31,160)(32,161)(33,162)(34,163)(35,164)(36,165)(37,166)(38,167)(39,168)(40,169)(41,170)(42,171)(43,172)(44,173)(45,174)(46,175)(47,176)(48,89)(49,90)(50,91)(51,92)(52,93)(53,94)(54,95)(55,96)(56,97)(57,98)(58,99)(59,100)(60,101)(61,102)(62,103)(63,104)(64,105)(65,106)(66,107)(67,108)(68,109)(69,110)(70,111)(71,112)(72,113)(73,114)(74,115)(75,116)(76,117)(77,118)(78,119)(79,120)(80,121)(81,122)(82,123)(83,124)(84,125)(85,126)(86,127)(87,128)(88,129), (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([(1,130),(2,131),(3,132),(4,133),(5,134),(6,135),(7,136),(8,137),(9,138),(10,139),(11,140),(12,141),(13,142),(14,143),(15,144),(16,145),(17,146),(18,147),(19,148),(20,149),(21,150),(22,151),(23,152),(24,153),(25,154),(26,155),(27,156),(28,157),(29,158),(30,159),(31,160),(32,161),(33,162),(34,163),(35,164),(36,165),(37,166),(38,167),(39,168),(40,169),(41,170),(42,171),(43,172),(44,173),(45,174),(46,175),(47,176),(48,89),(49,90),(50,91),(51,92),(52,93),(53,94),(54,95),(55,96),(56,97),(57,98),(58,99),(59,100),(60,101),(61,102),(62,103),(63,104),(64,105),(65,106),(66,107),(67,108),(68,109),(69,110),(70,111),(71,112),(72,113),(73,114),(74,115),(75,116),(76,117),(77,118),(78,119),(79,120),(80,121),(81,122),(82,123),(83,124),(84,125),(85,126),(86,127),(87,128),(88,129)], [(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)])

C2×C88 is a maximal subgroup of
C44.C8  Dic11⋊C8  C88⋊C4  C44.44D4  C44.4Q8  C44.5Q8  C88.C4  D22⋊C8  C2.D88  D44.2C4  D887C2

176 conjugacy classes

class 1 2A2B2C4A4B4C4D8A···8H11A···11J22A···22AD44A···44AN88A···88CB
order122244448···811···1122···2244···4488···88
size111111111···11···11···11···11···1

176 irreducible representations

dim111111111111
type+++
imageC1C2C2C4C4C8C11C22C22C44C44C88
kernelC2×C88C88C2×C44C44C2×C22C22C2×C8C8C2×C4C4C22C2
# reps121228102010202080

Matrix representation of C2×C88 in GL2(𝔽89) generated by

880
088
,
750
068
G:=sub<GL(2,GF(89))| [88,0,0,88],[75,0,0,68] >;

C2×C88 in GAP, Magma, Sage, TeX

C_2\times C_{88}
% in TeX

G:=Group("C2xC88");
// GroupNames label

G:=SmallGroup(176,22);
// by ID

G=gap.SmallGroup(176,22);
# by ID

G:=PCGroup([5,-2,-2,-11,-2,-2,220,58]);
// Polycyclic

G:=Group<a,b|a^2=b^88=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C2×C88 in TeX

׿
×
𝔽