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G = C2×C124order 248 = 23·31

Abelian group of type [2,124]

direct product, abelian, monomial, 2-elementary

Aliases: C2×C124, SmallGroup(248,8)

Series: Derived Chief Lower central Upper central

C1 — C2×C124
C1C2C62C124 — C2×C124
C1 — C2×C124
C1 — C2×C124

Generators and relations for C2×C124
 G = < a,b | a2=b124=1, ab=ba >


Smallest permutation representation of C2×C124
Regular action on 248 points
Generators in S248
(1 128)(2 129)(3 130)(4 131)(5 132)(6 133)(7 134)(8 135)(9 136)(10 137)(11 138)(12 139)(13 140)(14 141)(15 142)(16 143)(17 144)(18 145)(19 146)(20 147)(21 148)(22 149)(23 150)(24 151)(25 152)(26 153)(27 154)(28 155)(29 156)(30 157)(31 158)(32 159)(33 160)(34 161)(35 162)(36 163)(37 164)(38 165)(39 166)(40 167)(41 168)(42 169)(43 170)(44 171)(45 172)(46 173)(47 174)(48 175)(49 176)(50 177)(51 178)(52 179)(53 180)(54 181)(55 182)(56 183)(57 184)(58 185)(59 186)(60 187)(61 188)(62 189)(63 190)(64 191)(65 192)(66 193)(67 194)(68 195)(69 196)(70 197)(71 198)(72 199)(73 200)(74 201)(75 202)(76 203)(77 204)(78 205)(79 206)(80 207)(81 208)(82 209)(83 210)(84 211)(85 212)(86 213)(87 214)(88 215)(89 216)(90 217)(91 218)(92 219)(93 220)(94 221)(95 222)(96 223)(97 224)(98 225)(99 226)(100 227)(101 228)(102 229)(103 230)(104 231)(105 232)(106 233)(107 234)(108 235)(109 236)(110 237)(111 238)(112 239)(113 240)(114 241)(115 242)(116 243)(117 244)(118 245)(119 246)(120 247)(121 248)(122 125)(123 126)(124 127)
(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)

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

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

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

C2×C124 is a maximal subgroup of   C4.Dic31  Dic31⋊C4  C4⋊Dic31  D62⋊C4  D1245C2

248 conjugacy classes

class 1 2A2B2C4A4B4C4D31A···31AD62A···62CL124A···124DP
order1222444431···3162···62124···124
size111111111···11···11···1

248 irreducible representations

dim11111111
type+++
imageC1C2C2C4C31C62C62C124
kernelC2×C124C124C2×C62C62C2×C4C4C22C2
# reps1214306030120

Matrix representation of C2×C124 in GL2(𝔽373) generated by

3720
0372
,
2540
0142
G:=sub<GL(2,GF(373))| [372,0,0,372],[254,0,0,142] >;

C2×C124 in GAP, Magma, Sage, TeX

C_2\times C_{124}
% in TeX

G:=Group("C2xC124");
// GroupNames label

G:=SmallGroup(248,8);
// by ID

G=gap.SmallGroup(248,8);
# by ID

G:=PCGroup([4,-2,-2,-31,-2,496]);
// Polycyclic

G:=Group<a,b|a^2=b^124=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C2×C124 in TeX

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