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G = C4×C88order 352 = 25·11

Abelian group of type [4,88]

direct product, abelian, monomial, 2-elementary

Aliases: C4×C88, SmallGroup(352,45)

Series: Derived Chief Lower central Upper central

C1 — C4×C88
C1C2C22C2×C4C2×C44C2×C88 — C4×C88
C1 — C4×C88
C1 — C4×C88

Generators and relations for C4×C88
 G = < a,b | a4=b88=1, ab=ba >


Smallest permutation representation of C4×C88
Regular action on 352 points
Generators in S352
(1 152 235 325)(2 153 236 326)(3 154 237 327)(4 155 238 328)(5 156 239 329)(6 157 240 330)(7 158 241 331)(8 159 242 332)(9 160 243 333)(10 161 244 334)(11 162 245 335)(12 163 246 336)(13 164 247 337)(14 165 248 338)(15 166 249 339)(16 167 250 340)(17 168 251 341)(18 169 252 342)(19 170 253 343)(20 171 254 344)(21 172 255 345)(22 173 256 346)(23 174 257 347)(24 175 258 348)(25 176 259 349)(26 89 260 350)(27 90 261 351)(28 91 262 352)(29 92 263 265)(30 93 264 266)(31 94 177 267)(32 95 178 268)(33 96 179 269)(34 97 180 270)(35 98 181 271)(36 99 182 272)(37 100 183 273)(38 101 184 274)(39 102 185 275)(40 103 186 276)(41 104 187 277)(42 105 188 278)(43 106 189 279)(44 107 190 280)(45 108 191 281)(46 109 192 282)(47 110 193 283)(48 111 194 284)(49 112 195 285)(50 113 196 286)(51 114 197 287)(52 115 198 288)(53 116 199 289)(54 117 200 290)(55 118 201 291)(56 119 202 292)(57 120 203 293)(58 121 204 294)(59 122 205 295)(60 123 206 296)(61 124 207 297)(62 125 208 298)(63 126 209 299)(64 127 210 300)(65 128 211 301)(66 129 212 302)(67 130 213 303)(68 131 214 304)(69 132 215 305)(70 133 216 306)(71 134 217 307)(72 135 218 308)(73 136 219 309)(74 137 220 310)(75 138 221 311)(76 139 222 312)(77 140 223 313)(78 141 224 314)(79 142 225 315)(80 143 226 316)(81 144 227 317)(82 145 228 318)(83 146 229 319)(84 147 230 320)(85 148 231 321)(86 149 232 322)(87 150 233 323)(88 151 234 324)
(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)(265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352)

G:=sub<Sym(352)| (1,152,235,325)(2,153,236,326)(3,154,237,327)(4,155,238,328)(5,156,239,329)(6,157,240,330)(7,158,241,331)(8,159,242,332)(9,160,243,333)(10,161,244,334)(11,162,245,335)(12,163,246,336)(13,164,247,337)(14,165,248,338)(15,166,249,339)(16,167,250,340)(17,168,251,341)(18,169,252,342)(19,170,253,343)(20,171,254,344)(21,172,255,345)(22,173,256,346)(23,174,257,347)(24,175,258,348)(25,176,259,349)(26,89,260,350)(27,90,261,351)(28,91,262,352)(29,92,263,265)(30,93,264,266)(31,94,177,267)(32,95,178,268)(33,96,179,269)(34,97,180,270)(35,98,181,271)(36,99,182,272)(37,100,183,273)(38,101,184,274)(39,102,185,275)(40,103,186,276)(41,104,187,277)(42,105,188,278)(43,106,189,279)(44,107,190,280)(45,108,191,281)(46,109,192,282)(47,110,193,283)(48,111,194,284)(49,112,195,285)(50,113,196,286)(51,114,197,287)(52,115,198,288)(53,116,199,289)(54,117,200,290)(55,118,201,291)(56,119,202,292)(57,120,203,293)(58,121,204,294)(59,122,205,295)(60,123,206,296)(61,124,207,297)(62,125,208,298)(63,126,209,299)(64,127,210,300)(65,128,211,301)(66,129,212,302)(67,130,213,303)(68,131,214,304)(69,132,215,305)(70,133,216,306)(71,134,217,307)(72,135,218,308)(73,136,219,309)(74,137,220,310)(75,138,221,311)(76,139,222,312)(77,140,223,313)(78,141,224,314)(79,142,225,315)(80,143,226,316)(81,144,227,317)(82,145,228,318)(83,146,229,319)(84,147,230,320)(85,148,231,321)(86,149,232,322)(87,150,233,323)(88,151,234,324), (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)(265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352)>;

G:=Group( (1,152,235,325)(2,153,236,326)(3,154,237,327)(4,155,238,328)(5,156,239,329)(6,157,240,330)(7,158,241,331)(8,159,242,332)(9,160,243,333)(10,161,244,334)(11,162,245,335)(12,163,246,336)(13,164,247,337)(14,165,248,338)(15,166,249,339)(16,167,250,340)(17,168,251,341)(18,169,252,342)(19,170,253,343)(20,171,254,344)(21,172,255,345)(22,173,256,346)(23,174,257,347)(24,175,258,348)(25,176,259,349)(26,89,260,350)(27,90,261,351)(28,91,262,352)(29,92,263,265)(30,93,264,266)(31,94,177,267)(32,95,178,268)(33,96,179,269)(34,97,180,270)(35,98,181,271)(36,99,182,272)(37,100,183,273)(38,101,184,274)(39,102,185,275)(40,103,186,276)(41,104,187,277)(42,105,188,278)(43,106,189,279)(44,107,190,280)(45,108,191,281)(46,109,192,282)(47,110,193,283)(48,111,194,284)(49,112,195,285)(50,113,196,286)(51,114,197,287)(52,115,198,288)(53,116,199,289)(54,117,200,290)(55,118,201,291)(56,119,202,292)(57,120,203,293)(58,121,204,294)(59,122,205,295)(60,123,206,296)(61,124,207,297)(62,125,208,298)(63,126,209,299)(64,127,210,300)(65,128,211,301)(66,129,212,302)(67,130,213,303)(68,131,214,304)(69,132,215,305)(70,133,216,306)(71,134,217,307)(72,135,218,308)(73,136,219,309)(74,137,220,310)(75,138,221,311)(76,139,222,312)(77,140,223,313)(78,141,224,314)(79,142,225,315)(80,143,226,316)(81,144,227,317)(82,145,228,318)(83,146,229,319)(84,147,230,320)(85,148,231,321)(86,149,232,322)(87,150,233,323)(88,151,234,324), (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)(265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352) );

G=PermutationGroup([(1,152,235,325),(2,153,236,326),(3,154,237,327),(4,155,238,328),(5,156,239,329),(6,157,240,330),(7,158,241,331),(8,159,242,332),(9,160,243,333),(10,161,244,334),(11,162,245,335),(12,163,246,336),(13,164,247,337),(14,165,248,338),(15,166,249,339),(16,167,250,340),(17,168,251,341),(18,169,252,342),(19,170,253,343),(20,171,254,344),(21,172,255,345),(22,173,256,346),(23,174,257,347),(24,175,258,348),(25,176,259,349),(26,89,260,350),(27,90,261,351),(28,91,262,352),(29,92,263,265),(30,93,264,266),(31,94,177,267),(32,95,178,268),(33,96,179,269),(34,97,180,270),(35,98,181,271),(36,99,182,272),(37,100,183,273),(38,101,184,274),(39,102,185,275),(40,103,186,276),(41,104,187,277),(42,105,188,278),(43,106,189,279),(44,107,190,280),(45,108,191,281),(46,109,192,282),(47,110,193,283),(48,111,194,284),(49,112,195,285),(50,113,196,286),(51,114,197,287),(52,115,198,288),(53,116,199,289),(54,117,200,290),(55,118,201,291),(56,119,202,292),(57,120,203,293),(58,121,204,294),(59,122,205,295),(60,123,206,296),(61,124,207,297),(62,125,208,298),(63,126,209,299),(64,127,210,300),(65,128,211,301),(66,129,212,302),(67,130,213,303),(68,131,214,304),(69,132,215,305),(70,133,216,306),(71,134,217,307),(72,135,218,308),(73,136,219,309),(74,137,220,310),(75,138,221,311),(76,139,222,312),(77,140,223,313),(78,141,224,314),(79,142,225,315),(80,143,226,316),(81,144,227,317),(82,145,228,318),(83,146,229,319),(84,147,230,320),(85,148,231,321),(86,149,232,322),(87,150,233,323),(88,151,234,324)], [(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),(265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352)])

352 conjugacy classes

class 1 2A2B2C4A···4L8A···8P11A···11J22A···22AD44A···44DP88A···88FD
order12224···48···811···1122···2244···4488···88
size11111···11···11···11···11···11···1

352 irreducible representations

dim111111111111
type+++
imageC1C2C2C4C4C8C11C22C22C44C44C88
kernelC4×C88C4×C44C2×C88C88C2×C44C44C4×C8C42C2×C8C8C2×C4C4
# reps11284161010208040160

Matrix representation of C4×C88 in GL2(𝔽89) generated by

550
055
,
40
076
G:=sub<GL(2,GF(89))| [55,0,0,55],[4,0,0,76] >;

C4×C88 in GAP, Magma, Sage, TeX

C_4\times C_{88}
% in TeX

G:=Group("C4xC88");
// GroupNames label

G:=SmallGroup(352,45);
// by ID

G=gap.SmallGroup(352,45);
# by ID

G:=PCGroup([6,-2,-2,-11,-2,-2,-2,264,535,117]);
// Polycyclic

G:=Group<a,b|a^4=b^88=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C4×C88 in TeX

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