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## G = C2×C148order 296 = 23·37

### Abelian group of type [2,148]

Aliases: C2×C148, SmallGroup(296,9)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C148
 Chief series C1 — C2 — C74 — C148 — C2×C148
 Lower central C1 — C2×C148
 Upper central C1 — C2×C148

Generators and relations for C2×C148
G = < a,b | a2=b148=1, ab=ba >

Smallest permutation representation of C2×C148
Regular action on 296 points
Generators in S296
(1 198)(2 199)(3 200)(4 201)(5 202)(6 203)(7 204)(8 205)(9 206)(10 207)(11 208)(12 209)(13 210)(14 211)(15 212)(16 213)(17 214)(18 215)(19 216)(20 217)(21 218)(22 219)(23 220)(24 221)(25 222)(26 223)(27 224)(28 225)(29 226)(30 227)(31 228)(32 229)(33 230)(34 231)(35 232)(36 233)(37 234)(38 235)(39 236)(40 237)(41 238)(42 239)(43 240)(44 241)(45 242)(46 243)(47 244)(48 245)(49 246)(50 247)(51 248)(52 249)(53 250)(54 251)(55 252)(56 253)(57 254)(58 255)(59 256)(60 257)(61 258)(62 259)(63 260)(64 261)(65 262)(66 263)(67 264)(68 265)(69 266)(70 267)(71 268)(72 269)(73 270)(74 271)(75 272)(76 273)(77 274)(78 275)(79 276)(80 277)(81 278)(82 279)(83 280)(84 281)(85 282)(86 283)(87 284)(88 285)(89 286)(90 287)(91 288)(92 289)(93 290)(94 291)(95 292)(96 293)(97 294)(98 295)(99 296)(100 149)(101 150)(102 151)(103 152)(104 153)(105 154)(106 155)(107 156)(108 157)(109 158)(110 159)(111 160)(112 161)(113 162)(114 163)(115 164)(116 165)(117 166)(118 167)(119 168)(120 169)(121 170)(122 171)(123 172)(124 173)(125 174)(126 175)(127 176)(128 177)(129 178)(130 179)(131 180)(132 181)(133 182)(134 183)(135 184)(136 185)(137 186)(138 187)(139 188)(140 189)(141 190)(142 191)(143 192)(144 193)(145 194)(146 195)(147 196)(148 197)
(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)(149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296)

G:=sub<Sym(296)| (1,198)(2,199)(3,200)(4,201)(5,202)(6,203)(7,204)(8,205)(9,206)(10,207)(11,208)(12,209)(13,210)(14,211)(15,212)(16,213)(17,214)(18,215)(19,216)(20,217)(21,218)(22,219)(23,220)(24,221)(25,222)(26,223)(27,224)(28,225)(29,226)(30,227)(31,228)(32,229)(33,230)(34,231)(35,232)(36,233)(37,234)(38,235)(39,236)(40,237)(41,238)(42,239)(43,240)(44,241)(45,242)(46,243)(47,244)(48,245)(49,246)(50,247)(51,248)(52,249)(53,250)(54,251)(55,252)(56,253)(57,254)(58,255)(59,256)(60,257)(61,258)(62,259)(63,260)(64,261)(65,262)(66,263)(67,264)(68,265)(69,266)(70,267)(71,268)(72,269)(73,270)(74,271)(75,272)(76,273)(77,274)(78,275)(79,276)(80,277)(81,278)(82,279)(83,280)(84,281)(85,282)(86,283)(87,284)(88,285)(89,286)(90,287)(91,288)(92,289)(93,290)(94,291)(95,292)(96,293)(97,294)(98,295)(99,296)(100,149)(101,150)(102,151)(103,152)(104,153)(105,154)(106,155)(107,156)(108,157)(109,158)(110,159)(111,160)(112,161)(113,162)(114,163)(115,164)(116,165)(117,166)(118,167)(119,168)(120,169)(121,170)(122,171)(123,172)(124,173)(125,174)(126,175)(127,176)(128,177)(129,178)(130,179)(131,180)(132,181)(133,182)(134,183)(135,184)(136,185)(137,186)(138,187)(139,188)(140,189)(141,190)(142,191)(143,192)(144,193)(145,194)(146,195)(147,196)(148,197), (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)(149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296)>;

G:=Group( (1,198)(2,199)(3,200)(4,201)(5,202)(6,203)(7,204)(8,205)(9,206)(10,207)(11,208)(12,209)(13,210)(14,211)(15,212)(16,213)(17,214)(18,215)(19,216)(20,217)(21,218)(22,219)(23,220)(24,221)(25,222)(26,223)(27,224)(28,225)(29,226)(30,227)(31,228)(32,229)(33,230)(34,231)(35,232)(36,233)(37,234)(38,235)(39,236)(40,237)(41,238)(42,239)(43,240)(44,241)(45,242)(46,243)(47,244)(48,245)(49,246)(50,247)(51,248)(52,249)(53,250)(54,251)(55,252)(56,253)(57,254)(58,255)(59,256)(60,257)(61,258)(62,259)(63,260)(64,261)(65,262)(66,263)(67,264)(68,265)(69,266)(70,267)(71,268)(72,269)(73,270)(74,271)(75,272)(76,273)(77,274)(78,275)(79,276)(80,277)(81,278)(82,279)(83,280)(84,281)(85,282)(86,283)(87,284)(88,285)(89,286)(90,287)(91,288)(92,289)(93,290)(94,291)(95,292)(96,293)(97,294)(98,295)(99,296)(100,149)(101,150)(102,151)(103,152)(104,153)(105,154)(106,155)(107,156)(108,157)(109,158)(110,159)(111,160)(112,161)(113,162)(114,163)(115,164)(116,165)(117,166)(118,167)(119,168)(120,169)(121,170)(122,171)(123,172)(124,173)(125,174)(126,175)(127,176)(128,177)(129,178)(130,179)(131,180)(132,181)(133,182)(134,183)(135,184)(136,185)(137,186)(138,187)(139,188)(140,189)(141,190)(142,191)(143,192)(144,193)(145,194)(146,195)(147,196)(148,197), (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)(149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296) );

G=PermutationGroup([(1,198),(2,199),(3,200),(4,201),(5,202),(6,203),(7,204),(8,205),(9,206),(10,207),(11,208),(12,209),(13,210),(14,211),(15,212),(16,213),(17,214),(18,215),(19,216),(20,217),(21,218),(22,219),(23,220),(24,221),(25,222),(26,223),(27,224),(28,225),(29,226),(30,227),(31,228),(32,229),(33,230),(34,231),(35,232),(36,233),(37,234),(38,235),(39,236),(40,237),(41,238),(42,239),(43,240),(44,241),(45,242),(46,243),(47,244),(48,245),(49,246),(50,247),(51,248),(52,249),(53,250),(54,251),(55,252),(56,253),(57,254),(58,255),(59,256),(60,257),(61,258),(62,259),(63,260),(64,261),(65,262),(66,263),(67,264),(68,265),(69,266),(70,267),(71,268),(72,269),(73,270),(74,271),(75,272),(76,273),(77,274),(78,275),(79,276),(80,277),(81,278),(82,279),(83,280),(84,281),(85,282),(86,283),(87,284),(88,285),(89,286),(90,287),(91,288),(92,289),(93,290),(94,291),(95,292),(96,293),(97,294),(98,295),(99,296),(100,149),(101,150),(102,151),(103,152),(104,153),(105,154),(106,155),(107,156),(108,157),(109,158),(110,159),(111,160),(112,161),(113,162),(114,163),(115,164),(116,165),(117,166),(118,167),(119,168),(120,169),(121,170),(122,171),(123,172),(124,173),(125,174),(126,175),(127,176),(128,177),(129,178),(130,179),(131,180),(132,181),(133,182),(134,183),(135,184),(136,185),(137,186),(138,187),(139,188),(140,189),(141,190),(142,191),(143,192),(144,193),(145,194),(146,195),(147,196),(148,197)], [(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),(149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296)])

296 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 37A ··· 37AJ 74A ··· 74DD 148A ··· 148EN order 1 2 2 2 4 4 4 4 37 ··· 37 74 ··· 74 148 ··· 148 size 1 1 1 1 1 1 1 1 1 ··· 1 1 ··· 1 1 ··· 1

296 irreducible representations

 dim 1 1 1 1 1 1 1 1 type + + + image C1 C2 C2 C4 C37 C74 C74 C148 kernel C2×C148 C148 C2×C74 C74 C2×C4 C4 C22 C2 # reps 1 2 1 4 36 72 36 144

Matrix representation of C2×C148 in GL2(𝔽149) generated by

 148 0 0 148
,
 66 0 0 86
G:=sub<GL(2,GF(149))| [148,0,0,148],[66,0,0,86] >;

C2×C148 in GAP, Magma, Sage, TeX

C_2\times C_{148}
% in TeX

G:=Group("C2xC148");
// GroupNames label

G:=SmallGroup(296,9);
// by ID

G=gap.SmallGroup(296,9);
# by ID

G:=PCGroup([4,-2,-2,-37,-2,592]);
// Polycyclic

G:=Group<a,b|a^2=b^148=1,a*b=b*a>;
// generators/relations

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