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G = C2×C134order 268 = 22·67

Abelian group of type [2,134]

direct product, abelian, monomial, 2-elementary

Aliases: C2×C134, SmallGroup(268,4)

Series: Derived Chief Lower central Upper central

C1 — C2×C134
C1C67C134 — C2×C134
C1 — C2×C134
C1 — C2×C134

Generators and relations for C2×C134
 G = < a,b | a2=b134=1, ab=ba >


Smallest permutation representation of C2×C134
Regular action on 268 points
Generators in S268
(1 227)(2 228)(3 229)(4 230)(5 231)(6 232)(7 233)(8 234)(9 235)(10 236)(11 237)(12 238)(13 239)(14 240)(15 241)(16 242)(17 243)(18 244)(19 245)(20 246)(21 247)(22 248)(23 249)(24 250)(25 251)(26 252)(27 253)(28 254)(29 255)(30 256)(31 257)(32 258)(33 259)(34 260)(35 261)(36 262)(37 263)(38 264)(39 265)(40 266)(41 267)(42 268)(43 135)(44 136)(45 137)(46 138)(47 139)(48 140)(49 141)(50 142)(51 143)(52 144)(53 145)(54 146)(55 147)(56 148)(57 149)(58 150)(59 151)(60 152)(61 153)(62 154)(63 155)(64 156)(65 157)(66 158)(67 159)(68 160)(69 161)(70 162)(71 163)(72 164)(73 165)(74 166)(75 167)(76 168)(77 169)(78 170)(79 171)(80 172)(81 173)(82 174)(83 175)(84 176)(85 177)(86 178)(87 179)(88 180)(89 181)(90 182)(91 183)(92 184)(93 185)(94 186)(95 187)(96 188)(97 189)(98 190)(99 191)(100 192)(101 193)(102 194)(103 195)(104 196)(105 197)(106 198)(107 199)(108 200)(109 201)(110 202)(111 203)(112 204)(113 205)(114 206)(115 207)(116 208)(117 209)(118 210)(119 211)(120 212)(121 213)(122 214)(123 215)(124 216)(125 217)(126 218)(127 219)(128 220)(129 221)(130 222)(131 223)(132 224)(133 225)(134 226)
(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)(135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268)

G:=sub<Sym(268)| (1,227)(2,228)(3,229)(4,230)(5,231)(6,232)(7,233)(8,234)(9,235)(10,236)(11,237)(12,238)(13,239)(14,240)(15,241)(16,242)(17,243)(18,244)(19,245)(20,246)(21,247)(22,248)(23,249)(24,250)(25,251)(26,252)(27,253)(28,254)(29,255)(30,256)(31,257)(32,258)(33,259)(34,260)(35,261)(36,262)(37,263)(38,264)(39,265)(40,266)(41,267)(42,268)(43,135)(44,136)(45,137)(46,138)(47,139)(48,140)(49,141)(50,142)(51,143)(52,144)(53,145)(54,146)(55,147)(56,148)(57,149)(58,150)(59,151)(60,152)(61,153)(62,154)(63,155)(64,156)(65,157)(66,158)(67,159)(68,160)(69,161)(70,162)(71,163)(72,164)(73,165)(74,166)(75,167)(76,168)(77,169)(78,170)(79,171)(80,172)(81,173)(82,174)(83,175)(84,176)(85,177)(86,178)(87,179)(88,180)(89,181)(90,182)(91,183)(92,184)(93,185)(94,186)(95,187)(96,188)(97,189)(98,190)(99,191)(100,192)(101,193)(102,194)(103,195)(104,196)(105,197)(106,198)(107,199)(108,200)(109,201)(110,202)(111,203)(112,204)(113,205)(114,206)(115,207)(116,208)(117,209)(118,210)(119,211)(120,212)(121,213)(122,214)(123,215)(124,216)(125,217)(126,218)(127,219)(128,220)(129,221)(130,222)(131,223)(132,224)(133,225)(134,226), (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)(135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268)>;

G:=Group( (1,227)(2,228)(3,229)(4,230)(5,231)(6,232)(7,233)(8,234)(9,235)(10,236)(11,237)(12,238)(13,239)(14,240)(15,241)(16,242)(17,243)(18,244)(19,245)(20,246)(21,247)(22,248)(23,249)(24,250)(25,251)(26,252)(27,253)(28,254)(29,255)(30,256)(31,257)(32,258)(33,259)(34,260)(35,261)(36,262)(37,263)(38,264)(39,265)(40,266)(41,267)(42,268)(43,135)(44,136)(45,137)(46,138)(47,139)(48,140)(49,141)(50,142)(51,143)(52,144)(53,145)(54,146)(55,147)(56,148)(57,149)(58,150)(59,151)(60,152)(61,153)(62,154)(63,155)(64,156)(65,157)(66,158)(67,159)(68,160)(69,161)(70,162)(71,163)(72,164)(73,165)(74,166)(75,167)(76,168)(77,169)(78,170)(79,171)(80,172)(81,173)(82,174)(83,175)(84,176)(85,177)(86,178)(87,179)(88,180)(89,181)(90,182)(91,183)(92,184)(93,185)(94,186)(95,187)(96,188)(97,189)(98,190)(99,191)(100,192)(101,193)(102,194)(103,195)(104,196)(105,197)(106,198)(107,199)(108,200)(109,201)(110,202)(111,203)(112,204)(113,205)(114,206)(115,207)(116,208)(117,209)(118,210)(119,211)(120,212)(121,213)(122,214)(123,215)(124,216)(125,217)(126,218)(127,219)(128,220)(129,221)(130,222)(131,223)(132,224)(133,225)(134,226), (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)(135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268) );

G=PermutationGroup([(1,227),(2,228),(3,229),(4,230),(5,231),(6,232),(7,233),(8,234),(9,235),(10,236),(11,237),(12,238),(13,239),(14,240),(15,241),(16,242),(17,243),(18,244),(19,245),(20,246),(21,247),(22,248),(23,249),(24,250),(25,251),(26,252),(27,253),(28,254),(29,255),(30,256),(31,257),(32,258),(33,259),(34,260),(35,261),(36,262),(37,263),(38,264),(39,265),(40,266),(41,267),(42,268),(43,135),(44,136),(45,137),(46,138),(47,139),(48,140),(49,141),(50,142),(51,143),(52,144),(53,145),(54,146),(55,147),(56,148),(57,149),(58,150),(59,151),(60,152),(61,153),(62,154),(63,155),(64,156),(65,157),(66,158),(67,159),(68,160),(69,161),(70,162),(71,163),(72,164),(73,165),(74,166),(75,167),(76,168),(77,169),(78,170),(79,171),(80,172),(81,173),(82,174),(83,175),(84,176),(85,177),(86,178),(87,179),(88,180),(89,181),(90,182),(91,183),(92,184),(93,185),(94,186),(95,187),(96,188),(97,189),(98,190),(99,191),(100,192),(101,193),(102,194),(103,195),(104,196),(105,197),(106,198),(107,199),(108,200),(109,201),(110,202),(111,203),(112,204),(113,205),(114,206),(115,207),(116,208),(117,209),(118,210),(119,211),(120,212),(121,213),(122,214),(123,215),(124,216),(125,217),(126,218),(127,219),(128,220),(129,221),(130,222),(131,223),(132,224),(133,225),(134,226)], [(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),(135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268)])

268 conjugacy classes

class 1 2A2B2C67A···67BN134A···134GP
order122267···67134···134
size11111···11···1

268 irreducible representations

dim1111
type++
imageC1C2C67C134
kernelC2×C134C134C22C2
# reps1366198

Matrix representation of C2×C134 in GL2(𝔽269) generated by

2680
0268
,
1500
0256
G:=sub<GL(2,GF(269))| [268,0,0,268],[150,0,0,256] >;

C2×C134 in GAP, Magma, Sage, TeX

C_2\times C_{134}
% in TeX

G:=Group("C2xC134");
// GroupNames label

G:=SmallGroup(268,4);
// by ID

G=gap.SmallGroup(268,4);
# by ID

G:=PCGroup([3,-2,-2,-67]);
// Polycyclic

G:=Group<a,b|a^2=b^134=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C2×C134 in TeX

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