metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C88⋊4C4, C8⋊3Dic11, C22.4C42, C22.2M4(2), C11⋊C8⋊4C4, (C2×C8).8D11, C11⋊2(C8⋊C4), (C2×C88).12C2, C44.41(C2×C4), C4.21(C4×D11), (C2×C4).92D22, C2.4(C4×Dic11), C2.2(C88⋊C2), (C4×Dic11).6C2, (C2×Dic11).3C4, C4.13(C2×Dic11), C22.10(C4×D11), (C2×C44).106C22, (C2×C11⋊C8).10C2, (C2×C22).11(C2×C4), SmallGroup(352,21)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C88⋊C4
G = < a,b | a88=b4=1, bab-1=a21 >
Smallest permutation representation of C88⋊C4
►Regular action on 352 points
Generators in S352
(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)
(1 286 236 107)(2 307 237 128)(3 328 238 149)(4 349 239 170)(5 282 240 103)(6 303 241 124)(7 324 242 145)(8 345 243 166)(9 278 244 99)(10 299 245 120)(11 320 246 141)(12 341 247 162)(13 274 248 95)(14 295 249 116)(15 316 250 137)(16 337 251 158)(17 270 252 91)(18 291 253 112)(19 312 254 133)(20 333 255 154)(21 266 256 175)(22 287 257 108)(23 308 258 129)(24 329 259 150)(25 350 260 171)(26 283 261 104)(27 304 262 125)(28 325 263 146)(29 346 264 167)(30 279 177 100)(31 300 178 121)(32 321 179 142)(33 342 180 163)(34 275 181 96)(35 296 182 117)(36 317 183 138)(37 338 184 159)(38 271 185 92)(39 292 186 113)(40 313 187 134)(41 334 188 155)(42 267 189 176)(43 288 190 109)(44 309 191 130)(45 330 192 151)(46 351 193 172)(47 284 194 105)(48 305 195 126)(49 326 196 147)(50 347 197 168)(51 280 198 101)(52 301 199 122)(53 322 200 143)(54 343 201 164)(55 276 202 97)(56 297 203 118)(57 318 204 139)(58 339 205 160)(59 272 206 93)(60 293 207 114)(61 314 208 135)(62 335 209 156)(63 268 210 89)(64 289 211 110)(65 310 212 131)(66 331 213 152)(67 352 214 173)(68 285 215 106)(69 306 216 127)(70 327 217 148)(71 348 218 169)(72 281 219 102)(73 302 220 123)(74 323 221 144)(75 344 222 165)(76 277 223 98)(77 298 224 119)(78 319 225 140)(79 340 226 161)(80 273 227 94)(81 294 228 115)(82 315 229 136)(83 336 230 157)(84 269 231 90)(85 290 232 111)(86 311 233 132)(87 332 234 153)(88 265 235 174)
G:=sub<Sym(352)| (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), (1,286,236,107)(2,307,237,128)(3,328,238,149)(4,349,239,170)(5,282,240,103)(6,303,241,124)(7,324,242,145)(8,345,243,166)(9,278,244,99)(10,299,245,120)(11,320,246,141)(12,341,247,162)(13,274,248,95)(14,295,249,116)(15,316,250,137)(16,337,251,158)(17,270,252,91)(18,291,253,112)(19,312,254,133)(20,333,255,154)(21,266,256,175)(22,287,257,108)(23,308,258,129)(24,329,259,150)(25,350,260,171)(26,283,261,104)(27,304,262,125)(28,325,263,146)(29,346,264,167)(30,279,177,100)(31,300,178,121)(32,321,179,142)(33,342,180,163)(34,275,181,96)(35,296,182,117)(36,317,183,138)(37,338,184,159)(38,271,185,92)(39,292,186,113)(40,313,187,134)(41,334,188,155)(42,267,189,176)(43,288,190,109)(44,309,191,130)(45,330,192,151)(46,351,193,172)(47,284,194,105)(48,305,195,126)(49,326,196,147)(50,347,197,168)(51,280,198,101)(52,301,199,122)(53,322,200,143)(54,343,201,164)(55,276,202,97)(56,297,203,118)(57,318,204,139)(58,339,205,160)(59,272,206,93)(60,293,207,114)(61,314,208,135)(62,335,209,156)(63,268,210,89)(64,289,211,110)(65,310,212,131)(66,331,213,152)(67,352,214,173)(68,285,215,106)(69,306,216,127)(70,327,217,148)(71,348,218,169)(72,281,219,102)(73,302,220,123)(74,323,221,144)(75,344,222,165)(76,277,223,98)(77,298,224,119)(78,319,225,140)(79,340,226,161)(80,273,227,94)(81,294,228,115)(82,315,229,136)(83,336,230,157)(84,269,231,90)(85,290,232,111)(86,311,233,132)(87,332,234,153)(88,265,235,174)>;
G:=Group( (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), (1,286,236,107)(2,307,237,128)(3,328,238,149)(4,349,239,170)(5,282,240,103)(6,303,241,124)(7,324,242,145)(8,345,243,166)(9,278,244,99)(10,299,245,120)(11,320,246,141)(12,341,247,162)(13,274,248,95)(14,295,249,116)(15,316,250,137)(16,337,251,158)(17,270,252,91)(18,291,253,112)(19,312,254,133)(20,333,255,154)(21,266,256,175)(22,287,257,108)(23,308,258,129)(24,329,259,150)(25,350,260,171)(26,283,261,104)(27,304,262,125)(28,325,263,146)(29,346,264,167)(30,279,177,100)(31,300,178,121)(32,321,179,142)(33,342,180,163)(34,275,181,96)(35,296,182,117)(36,317,183,138)(37,338,184,159)(38,271,185,92)(39,292,186,113)(40,313,187,134)(41,334,188,155)(42,267,189,176)(43,288,190,109)(44,309,191,130)(45,330,192,151)(46,351,193,172)(47,284,194,105)(48,305,195,126)(49,326,196,147)(50,347,197,168)(51,280,198,101)(52,301,199,122)(53,322,200,143)(54,343,201,164)(55,276,202,97)(56,297,203,118)(57,318,204,139)(58,339,205,160)(59,272,206,93)(60,293,207,114)(61,314,208,135)(62,335,209,156)(63,268,210,89)(64,289,211,110)(65,310,212,131)(66,331,213,152)(67,352,214,173)(68,285,215,106)(69,306,216,127)(70,327,217,148)(71,348,218,169)(72,281,219,102)(73,302,220,123)(74,323,221,144)(75,344,222,165)(76,277,223,98)(77,298,224,119)(78,319,225,140)(79,340,226,161)(80,273,227,94)(81,294,228,115)(82,315,229,136)(83,336,230,157)(84,269,231,90)(85,290,232,111)(86,311,233,132)(87,332,234,153)(88,265,235,174) );
G=PermutationGroup([[(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)], [(1,286,236,107),(2,307,237,128),(3,328,238,149),(4,349,239,170),(5,282,240,103),(6,303,241,124),(7,324,242,145),(8,345,243,166),(9,278,244,99),(10,299,245,120),(11,320,246,141),(12,341,247,162),(13,274,248,95),(14,295,249,116),(15,316,250,137),(16,337,251,158),(17,270,252,91),(18,291,253,112),(19,312,254,133),(20,333,255,154),(21,266,256,175),(22,287,257,108),(23,308,258,129),(24,329,259,150),(25,350,260,171),(26,283,261,104),(27,304,262,125),(28,325,263,146),(29,346,264,167),(30,279,177,100),(31,300,178,121),(32,321,179,142),(33,342,180,163),(34,275,181,96),(35,296,182,117),(36,317,183,138),(37,338,184,159),(38,271,185,92),(39,292,186,113),(40,313,187,134),(41,334,188,155),(42,267,189,176),(43,288,190,109),(44,309,191,130),(45,330,192,151),(46,351,193,172),(47,284,194,105),(48,305,195,126),(49,326,196,147),(50,347,197,168),(51,280,198,101),(52,301,199,122),(53,322,200,143),(54,343,201,164),(55,276,202,97),(56,297,203,118),(57,318,204,139),(58,339,205,160),(59,272,206,93),(60,293,207,114),(61,314,208,135),(62,335,209,156),(63,268,210,89),(64,289,211,110),(65,310,212,131),(66,331,213,152),(67,352,214,173),(68,285,215,106),(69,306,216,127),(70,327,217,148),(71,348,218,169),(72,281,219,102),(73,302,220,123),(74,323,221,144),(75,344,222,165),(76,277,223,98),(77,298,224,119),(78,319,225,140),(79,340,226,161),(80,273,227,94),(81,294,228,115),(82,315,229,136),(83,336,230,157),(84,269,231,90),(85,290,232,111),(86,311,233,132),(87,332,234,153),(88,265,235,174)]])
100 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 11A | ··· | 11E | 22A | ··· | 22O | 44A | ··· | 44T | 88A | ··· | 88AN |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 11 | ··· | 11 | 22 | ··· | 22 | 44 | ··· | 44 | 88 | ··· | 88 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 22 | 22 | 22 | 22 | 2 | 2 | 2 | 2 | 22 | 22 | 22 | 22 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
100 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | |||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | M4(2) | D11 | Dic11 | D22 | C4×D11 | C4×D11 | C88⋊C2 |
| kernel | C88⋊C4 | C2×C11⋊C8 | C4×Dic11 | C2×C88 | C11⋊C8 | C88 | C2×Dic11 | C22 | C2×C8 | C8 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 5 | 10 | 5 | 10 | 10 | 40 |
Matrix representation of C88⋊C4 ►in GL3(𝔽89) generated by
| 88 | 0 | 0 |
| 0 | 17 | 6 |
| 0 | 21 | 63 |
| 55 | 0 | 0 |
| 0 | 2 | 50 |
| 0 | 48 | 87 |
G:=sub<GL(3,GF(89))| [88,0,0,0,17,21,0,6,63],[55,0,0,0,2,48,0,50,87] >;
C88⋊C4 in GAP, Magma, Sage, TeX
C_{88}\rtimes C_4 % in TeX
G:=Group("C88:C4"); // GroupNames label
G:=SmallGroup(352,21);
// by ID
G=gap.SmallGroup(352,21);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-11,24,217,55,69,11525]);
// Polycyclic
G:=Group<a,b|a^88=b^4=1,b*a*b^-1=a^21>;
// generators/relations
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