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G = C2×C96order 192 = 26·3

Abelian group of type [2,96]

direct product, abelian, monomial, 2-elementary

Aliases: C2×C96, SmallGroup(192,175)

Series: Derived Chief Lower central Upper central

C1 — C2×C96
C1C2C4C8C16C48C96 — C2×C96
C1 — C2×C96
C1 — C2×C96

Generators and relations for C2×C96
 G = < a,b | a2=b96=1, ab=ba >


Smallest permutation representation of C2×C96
Regular action on 192 points
Generators in S192
(1 164)(2 165)(3 166)(4 167)(5 168)(6 169)(7 170)(8 171)(9 172)(10 173)(11 174)(12 175)(13 176)(14 177)(15 178)(16 179)(17 180)(18 181)(19 182)(20 183)(21 184)(22 185)(23 186)(24 187)(25 188)(26 189)(27 190)(28 191)(29 192)(30 97)(31 98)(32 99)(33 100)(34 101)(35 102)(36 103)(37 104)(38 105)(39 106)(40 107)(41 108)(42 109)(43 110)(44 111)(45 112)(46 113)(47 114)(48 115)(49 116)(50 117)(51 118)(52 119)(53 120)(54 121)(55 122)(56 123)(57 124)(58 125)(59 126)(60 127)(61 128)(62 129)(63 130)(64 131)(65 132)(66 133)(67 134)(68 135)(69 136)(70 137)(71 138)(72 139)(73 140)(74 141)(75 142)(76 143)(77 144)(78 145)(79 146)(80 147)(81 148)(82 149)(83 150)(84 151)(85 152)(86 153)(87 154)(88 155)(89 156)(90 157)(91 158)(92 159)(93 160)(94 161)(95 162)(96 163)
(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192)

G:=sub<Sym(192)| (1,164)(2,165)(3,166)(4,167)(5,168)(6,169)(7,170)(8,171)(9,172)(10,173)(11,174)(12,175)(13,176)(14,177)(15,178)(16,179)(17,180)(18,181)(19,182)(20,183)(21,184)(22,185)(23,186)(24,187)(25,188)(26,189)(27,190)(28,191)(29,192)(30,97)(31,98)(32,99)(33,100)(34,101)(35,102)(36,103)(37,104)(38,105)(39,106)(40,107)(41,108)(42,109)(43,110)(44,111)(45,112)(46,113)(47,114)(48,115)(49,116)(50,117)(51,118)(52,119)(53,120)(54,121)(55,122)(56,123)(57,124)(58,125)(59,126)(60,127)(61,128)(62,129)(63,130)(64,131)(65,132)(66,133)(67,134)(68,135)(69,136)(70,137)(71,138)(72,139)(73,140)(74,141)(75,142)(76,143)(77,144)(78,145)(79,146)(80,147)(81,148)(82,149)(83,150)(84,151)(85,152)(86,153)(87,154)(88,155)(89,156)(90,157)(91,158)(92,159)(93,160)(94,161)(95,162)(96,163), (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)(97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192)>;

G:=Group( (1,164)(2,165)(3,166)(4,167)(5,168)(6,169)(7,170)(8,171)(9,172)(10,173)(11,174)(12,175)(13,176)(14,177)(15,178)(16,179)(17,180)(18,181)(19,182)(20,183)(21,184)(22,185)(23,186)(24,187)(25,188)(26,189)(27,190)(28,191)(29,192)(30,97)(31,98)(32,99)(33,100)(34,101)(35,102)(36,103)(37,104)(38,105)(39,106)(40,107)(41,108)(42,109)(43,110)(44,111)(45,112)(46,113)(47,114)(48,115)(49,116)(50,117)(51,118)(52,119)(53,120)(54,121)(55,122)(56,123)(57,124)(58,125)(59,126)(60,127)(61,128)(62,129)(63,130)(64,131)(65,132)(66,133)(67,134)(68,135)(69,136)(70,137)(71,138)(72,139)(73,140)(74,141)(75,142)(76,143)(77,144)(78,145)(79,146)(80,147)(81,148)(82,149)(83,150)(84,151)(85,152)(86,153)(87,154)(88,155)(89,156)(90,157)(91,158)(92,159)(93,160)(94,161)(95,162)(96,163), (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)(97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192) );

G=PermutationGroup([(1,164),(2,165),(3,166),(4,167),(5,168),(6,169),(7,170),(8,171),(9,172),(10,173),(11,174),(12,175),(13,176),(14,177),(15,178),(16,179),(17,180),(18,181),(19,182),(20,183),(21,184),(22,185),(23,186),(24,187),(25,188),(26,189),(27,190),(28,191),(29,192),(30,97),(31,98),(32,99),(33,100),(34,101),(35,102),(36,103),(37,104),(38,105),(39,106),(40,107),(41,108),(42,109),(43,110),(44,111),(45,112),(46,113),(47,114),(48,115),(49,116),(50,117),(51,118),(52,119),(53,120),(54,121),(55,122),(56,123),(57,124),(58,125),(59,126),(60,127),(61,128),(62,129),(63,130),(64,131),(65,132),(66,133),(67,134),(68,135),(69,136),(70,137),(71,138),(72,139),(73,140),(74,141),(75,142),(76,143),(77,144),(78,145),(79,146),(80,147),(81,148),(82,149),(83,150),(84,151),(85,152),(86,153),(87,154),(88,155),(89,156),(90,157),(91,158),(92,159),(93,160),(94,161),(95,162),(96,163)], [(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),(97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192)])

192 conjugacy classes

class 1 2A2B2C3A3B4A4B4C4D6A···6F8A···8H12A···12H16A···16P24A···24P32A···32AF48A···48AF96A···96BL
order12223344446···68···812···1216···1624···2432···3248···4896···96
size11111111111···11···11···11···11···11···11···11···1

192 irreducible representations

dim11111111111111111111
type+++
imageC1C2C2C3C4C4C6C6C8C8C12C12C16C16C24C24C32C48C48C96
kernelC2×C96C96C2×C48C2×C32C48C2×C24C32C2×C16C24C2×C12C16C2×C8C12C2×C6C8C2×C4C6C4C22C2
# reps121222424444888832161664

Matrix representation of C2×C96 in GL2(𝔽97) generated by

10
096
,
100
093
G:=sub<GL(2,GF(97))| [1,0,0,96],[10,0,0,93] >;

C2×C96 in GAP, Magma, Sage, TeX

C_2\times C_{96}
% in TeX

G:=Group("C2xC96");
// GroupNames label

G:=SmallGroup(192,175);
// by ID

G=gap.SmallGroup(192,175);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-2,84,80,102,124]);
// Polycyclic

G:=Group<a,b|a^2=b^96=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C2×C96 in TeX

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