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G = C2×C126order 252 = 22·32·7

Abelian group of type [2,126]

direct product, abelian, monomial, 2-elementary

Aliases: C2×C126, SmallGroup(252,15)

Series: Derived Chief Lower central Upper central

C1 — C2×C126
C1C3C21C63C126 — C2×C126
C1 — C2×C126
C1 — C2×C126

Generators and relations for C2×C126
 G = < a,b | a2=b126=1, ab=ba >


Smallest permutation representation of C2×C126
Regular action on 252 points
Generators in S252
(1 127)(2 128)(3 129)(4 130)(5 131)(6 132)(7 133)(8 134)(9 135)(10 136)(11 137)(12 138)(13 139)(14 140)(15 141)(16 142)(17 143)(18 144)(19 145)(20 146)(21 147)(22 148)(23 149)(24 150)(25 151)(26 152)(27 153)(28 154)(29 155)(30 156)(31 157)(32 158)(33 159)(34 160)(35 161)(36 162)(37 163)(38 164)(39 165)(40 166)(41 167)(42 168)(43 169)(44 170)(45 171)(46 172)(47 173)(48 174)(49 175)(50 176)(51 177)(52 178)(53 179)(54 180)(55 181)(56 182)(57 183)(58 184)(59 185)(60 186)(61 187)(62 188)(63 189)(64 190)(65 191)(66 192)(67 193)(68 194)(69 195)(70 196)(71 197)(72 198)(73 199)(74 200)(75 201)(76 202)(77 203)(78 204)(79 205)(80 206)(81 207)(82 208)(83 209)(84 210)(85 211)(86 212)(87 213)(88 214)(89 215)(90 216)(91 217)(92 218)(93 219)(94 220)(95 221)(96 222)(97 223)(98 224)(99 225)(100 226)(101 227)(102 228)(103 229)(104 230)(105 231)(106 232)(107 233)(108 234)(109 235)(110 236)(111 237)(112 238)(113 239)(114 240)(115 241)(116 242)(117 243)(118 244)(119 245)(120 246)(121 247)(122 248)(123 249)(124 250)(125 251)(126 252)
(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)(127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252)

G:=sub<Sym(252)| (1,127)(2,128)(3,129)(4,130)(5,131)(6,132)(7,133)(8,134)(9,135)(10,136)(11,137)(12,138)(13,139)(14,140)(15,141)(16,142)(17,143)(18,144)(19,145)(20,146)(21,147)(22,148)(23,149)(24,150)(25,151)(26,152)(27,153)(28,154)(29,155)(30,156)(31,157)(32,158)(33,159)(34,160)(35,161)(36,162)(37,163)(38,164)(39,165)(40,166)(41,167)(42,168)(43,169)(44,170)(45,171)(46,172)(47,173)(48,174)(49,175)(50,176)(51,177)(52,178)(53,179)(54,180)(55,181)(56,182)(57,183)(58,184)(59,185)(60,186)(61,187)(62,188)(63,189)(64,190)(65,191)(66,192)(67,193)(68,194)(69,195)(70,196)(71,197)(72,198)(73,199)(74,200)(75,201)(76,202)(77,203)(78,204)(79,205)(80,206)(81,207)(82,208)(83,209)(84,210)(85,211)(86,212)(87,213)(88,214)(89,215)(90,216)(91,217)(92,218)(93,219)(94,220)(95,221)(96,222)(97,223)(98,224)(99,225)(100,226)(101,227)(102,228)(103,229)(104,230)(105,231)(106,232)(107,233)(108,234)(109,235)(110,236)(111,237)(112,238)(113,239)(114,240)(115,241)(116,242)(117,243)(118,244)(119,245)(120,246)(121,247)(122,248)(123,249)(124,250)(125,251)(126,252), (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)(127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252)>;

G:=Group( (1,127)(2,128)(3,129)(4,130)(5,131)(6,132)(7,133)(8,134)(9,135)(10,136)(11,137)(12,138)(13,139)(14,140)(15,141)(16,142)(17,143)(18,144)(19,145)(20,146)(21,147)(22,148)(23,149)(24,150)(25,151)(26,152)(27,153)(28,154)(29,155)(30,156)(31,157)(32,158)(33,159)(34,160)(35,161)(36,162)(37,163)(38,164)(39,165)(40,166)(41,167)(42,168)(43,169)(44,170)(45,171)(46,172)(47,173)(48,174)(49,175)(50,176)(51,177)(52,178)(53,179)(54,180)(55,181)(56,182)(57,183)(58,184)(59,185)(60,186)(61,187)(62,188)(63,189)(64,190)(65,191)(66,192)(67,193)(68,194)(69,195)(70,196)(71,197)(72,198)(73,199)(74,200)(75,201)(76,202)(77,203)(78,204)(79,205)(80,206)(81,207)(82,208)(83,209)(84,210)(85,211)(86,212)(87,213)(88,214)(89,215)(90,216)(91,217)(92,218)(93,219)(94,220)(95,221)(96,222)(97,223)(98,224)(99,225)(100,226)(101,227)(102,228)(103,229)(104,230)(105,231)(106,232)(107,233)(108,234)(109,235)(110,236)(111,237)(112,238)(113,239)(114,240)(115,241)(116,242)(117,243)(118,244)(119,245)(120,246)(121,247)(122,248)(123,249)(124,250)(125,251)(126,252), (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)(127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252) );

G=PermutationGroup([(1,127),(2,128),(3,129),(4,130),(5,131),(6,132),(7,133),(8,134),(9,135),(10,136),(11,137),(12,138),(13,139),(14,140),(15,141),(16,142),(17,143),(18,144),(19,145),(20,146),(21,147),(22,148),(23,149),(24,150),(25,151),(26,152),(27,153),(28,154),(29,155),(30,156),(31,157),(32,158),(33,159),(34,160),(35,161),(36,162),(37,163),(38,164),(39,165),(40,166),(41,167),(42,168),(43,169),(44,170),(45,171),(46,172),(47,173),(48,174),(49,175),(50,176),(51,177),(52,178),(53,179),(54,180),(55,181),(56,182),(57,183),(58,184),(59,185),(60,186),(61,187),(62,188),(63,189),(64,190),(65,191),(66,192),(67,193),(68,194),(69,195),(70,196),(71,197),(72,198),(73,199),(74,200),(75,201),(76,202),(77,203),(78,204),(79,205),(80,206),(81,207),(82,208),(83,209),(84,210),(85,211),(86,212),(87,213),(88,214),(89,215),(90,216),(91,217),(92,218),(93,219),(94,220),(95,221),(96,222),(97,223),(98,224),(99,225),(100,226),(101,227),(102,228),(103,229),(104,230),(105,231),(106,232),(107,233),(108,234),(109,235),(110,236),(111,237),(112,238),(113,239),(114,240),(115,241),(116,242),(117,243),(118,244),(119,245),(120,246),(121,247),(122,248),(123,249),(124,250),(125,251),(126,252)], [(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),(127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252)])

252 conjugacy classes

class 1 2A2B2C3A3B6A···6F7A···7F9A···9F14A···14R18A···18R21A···21L42A···42AJ63A···63AJ126A···126DD
order1222336···67···79···914···1418···1821···2142···4263···63126···126
size1111111···11···11···11···11···11···11···11···11···1

252 irreducible representations

dim111111111111
type++
imageC1C2C3C6C7C9C14C18C21C42C63C126
kernelC2×C126C126C2×C42C42C2×C18C2×C14C18C14C2×C6C6C22C2
# reps1326661818123636108

Matrix representation of C2×C126 in GL2(𝔽127) generated by

10
0126
,
1010
099
G:=sub<GL(2,GF(127))| [1,0,0,126],[101,0,0,99] >;

C2×C126 in GAP, Magma, Sage, TeX

C_2\times C_{126}
% in TeX

G:=Group("C2xC126");
// GroupNames label

G:=SmallGroup(252,15);
// by ID

G=gap.SmallGroup(252,15);
# by ID

G:=PCGroup([5,-2,-2,-3,-7,-3,327]);
// Polycyclic

G:=Group<a,b|a^2=b^126=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C2×C126 in TeX

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