metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C44⋊1C8, C44.7Q8, C44.32D4, C4.16D44, C4.7Dic22, C42.2D11, C22.5M4(2), C4⋊(C11⋊C8), C11⋊1(C4⋊C8), (C2×C44).7C4, (C4×C44).4C2, C22.7(C2×C8), C22.1(C4⋊C4), (C2×C4).89D22, (C2×C4).3Dic11, C2.1(C44⋊C4), C2.2(C44.C4), (C2×C44).103C22, C22.8(C2×Dic11), C2.3(C2×C11⋊C8), (C2×C11⋊C8).8C2, (C2×C22).26(C2×C4), SmallGroup(352,10)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C44⋊C8
G = < a,b | a44=b8=1, bab-1=a-1 >
Smallest permutation representation of C44⋊C8
►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 78 201 161 239 349 127 274)(2 77 202 160 240 348 128 273)(3 76 203 159 241 347 129 272)(4 75 204 158 242 346 130 271)(5 74 205 157 243 345 131 270)(6 73 206 156 244 344 132 269)(7 72 207 155 245 343 89 268)(8 71 208 154 246 342 90 267)(9 70 209 153 247 341 91 266)(10 69 210 152 248 340 92 265)(11 68 211 151 249 339 93 308)(12 67 212 150 250 338 94 307)(13 66 213 149 251 337 95 306)(14 65 214 148 252 336 96 305)(15 64 215 147 253 335 97 304)(16 63 216 146 254 334 98 303)(17 62 217 145 255 333 99 302)(18 61 218 144 256 332 100 301)(19 60 219 143 257 331 101 300)(20 59 220 142 258 330 102 299)(21 58 177 141 259 329 103 298)(22 57 178 140 260 328 104 297)(23 56 179 139 261 327 105 296)(24 55 180 138 262 326 106 295)(25 54 181 137 263 325 107 294)(26 53 182 136 264 324 108 293)(27 52 183 135 221 323 109 292)(28 51 184 134 222 322 110 291)(29 50 185 133 223 321 111 290)(30 49 186 176 224 320 112 289)(31 48 187 175 225 319 113 288)(32 47 188 174 226 318 114 287)(33 46 189 173 227 317 115 286)(34 45 190 172 228 316 116 285)(35 88 191 171 229 315 117 284)(36 87 192 170 230 314 118 283)(37 86 193 169 231 313 119 282)(38 85 194 168 232 312 120 281)(39 84 195 167 233 311 121 280)(40 83 196 166 234 310 122 279)(41 82 197 165 235 309 123 278)(42 81 198 164 236 352 124 277)(43 80 199 163 237 351 125 276)(44 79 200 162 238 350 126 275)
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), (1,78,201,161,239,349,127,274)(2,77,202,160,240,348,128,273)(3,76,203,159,241,347,129,272)(4,75,204,158,242,346,130,271)(5,74,205,157,243,345,131,270)(6,73,206,156,244,344,132,269)(7,72,207,155,245,343,89,268)(8,71,208,154,246,342,90,267)(9,70,209,153,247,341,91,266)(10,69,210,152,248,340,92,265)(11,68,211,151,249,339,93,308)(12,67,212,150,250,338,94,307)(13,66,213,149,251,337,95,306)(14,65,214,148,252,336,96,305)(15,64,215,147,253,335,97,304)(16,63,216,146,254,334,98,303)(17,62,217,145,255,333,99,302)(18,61,218,144,256,332,100,301)(19,60,219,143,257,331,101,300)(20,59,220,142,258,330,102,299)(21,58,177,141,259,329,103,298)(22,57,178,140,260,328,104,297)(23,56,179,139,261,327,105,296)(24,55,180,138,262,326,106,295)(25,54,181,137,263,325,107,294)(26,53,182,136,264,324,108,293)(27,52,183,135,221,323,109,292)(28,51,184,134,222,322,110,291)(29,50,185,133,223,321,111,290)(30,49,186,176,224,320,112,289)(31,48,187,175,225,319,113,288)(32,47,188,174,226,318,114,287)(33,46,189,173,227,317,115,286)(34,45,190,172,228,316,116,285)(35,88,191,171,229,315,117,284)(36,87,192,170,230,314,118,283)(37,86,193,169,231,313,119,282)(38,85,194,168,232,312,120,281)(39,84,195,167,233,311,121,280)(40,83,196,166,234,310,122,279)(41,82,197,165,235,309,123,278)(42,81,198,164,236,352,124,277)(43,80,199,163,237,351,125,276)(44,79,200,162,238,350,126,275)>;
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), (1,78,201,161,239,349,127,274)(2,77,202,160,240,348,128,273)(3,76,203,159,241,347,129,272)(4,75,204,158,242,346,130,271)(5,74,205,157,243,345,131,270)(6,73,206,156,244,344,132,269)(7,72,207,155,245,343,89,268)(8,71,208,154,246,342,90,267)(9,70,209,153,247,341,91,266)(10,69,210,152,248,340,92,265)(11,68,211,151,249,339,93,308)(12,67,212,150,250,338,94,307)(13,66,213,149,251,337,95,306)(14,65,214,148,252,336,96,305)(15,64,215,147,253,335,97,304)(16,63,216,146,254,334,98,303)(17,62,217,145,255,333,99,302)(18,61,218,144,256,332,100,301)(19,60,219,143,257,331,101,300)(20,59,220,142,258,330,102,299)(21,58,177,141,259,329,103,298)(22,57,178,140,260,328,104,297)(23,56,179,139,261,327,105,296)(24,55,180,138,262,326,106,295)(25,54,181,137,263,325,107,294)(26,53,182,136,264,324,108,293)(27,52,183,135,221,323,109,292)(28,51,184,134,222,322,110,291)(29,50,185,133,223,321,111,290)(30,49,186,176,224,320,112,289)(31,48,187,175,225,319,113,288)(32,47,188,174,226,318,114,287)(33,46,189,173,227,317,115,286)(34,45,190,172,228,316,116,285)(35,88,191,171,229,315,117,284)(36,87,192,170,230,314,118,283)(37,86,193,169,231,313,119,282)(38,85,194,168,232,312,120,281)(39,84,195,167,233,311,121,280)(40,83,196,166,234,310,122,279)(41,82,197,165,235,309,123,278)(42,81,198,164,236,352,124,277)(43,80,199,163,237,351,125,276)(44,79,200,162,238,350,126,275) );
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)], [(1,78,201,161,239,349,127,274),(2,77,202,160,240,348,128,273),(3,76,203,159,241,347,129,272),(4,75,204,158,242,346,130,271),(5,74,205,157,243,345,131,270),(6,73,206,156,244,344,132,269),(7,72,207,155,245,343,89,268),(8,71,208,154,246,342,90,267),(9,70,209,153,247,341,91,266),(10,69,210,152,248,340,92,265),(11,68,211,151,249,339,93,308),(12,67,212,150,250,338,94,307),(13,66,213,149,251,337,95,306),(14,65,214,148,252,336,96,305),(15,64,215,147,253,335,97,304),(16,63,216,146,254,334,98,303),(17,62,217,145,255,333,99,302),(18,61,218,144,256,332,100,301),(19,60,219,143,257,331,101,300),(20,59,220,142,258,330,102,299),(21,58,177,141,259,329,103,298),(22,57,178,140,260,328,104,297),(23,56,179,139,261,327,105,296),(24,55,180,138,262,326,106,295),(25,54,181,137,263,325,107,294),(26,53,182,136,264,324,108,293),(27,52,183,135,221,323,109,292),(28,51,184,134,222,322,110,291),(29,50,185,133,223,321,111,290),(30,49,186,176,224,320,112,289),(31,48,187,175,225,319,113,288),(32,47,188,174,226,318,114,287),(33,46,189,173,227,317,115,286),(34,45,190,172,228,316,116,285),(35,88,191,171,229,315,117,284),(36,87,192,170,230,314,118,283),(37,86,193,169,231,313,119,282),(38,85,194,168,232,312,120,281),(39,84,195,167,233,311,121,280),(40,83,196,166,234,310,122,279),(41,82,197,165,235,309,123,278),(42,81,198,164,236,352,124,277),(43,80,199,163,237,351,125,276),(44,79,200,162,238,350,126,275)]])
100 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 8A | ··· | 8H | 11A | ··· | 11E | 22A | ··· | 22O | 44A | ··· | 44BH |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 11 | ··· | 11 | 22 | ··· | 22 | 44 | ··· | 44 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 22 | ··· | 22 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
100 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | + | - | + | |||||
| image | C1 | C2 | C2 | C4 | C8 | D4 | Q8 | M4(2) | D11 | Dic11 | D22 | C11⋊C8 | Dic22 | D44 | C44.C4 |
| kernel | C44⋊C8 | C2×C11⋊C8 | C4×C44 | C2×C44 | C44 | C44 | C44 | C22 | C42 | C2×C4 | C2×C4 | C4 | C4 | C4 | C2 |
| # reps | 1 | 2 | 1 | 4 | 8 | 1 | 1 | 2 | 5 | 10 | 5 | 20 | 10 | 10 | 20 |
Matrix representation of C44⋊C8 ►in GL3(𝔽89) generated by
| 1 | 0 | 0 |
| 0 | 69 | 0 |
| 0 | 54 | 40 |
| 37 | 0 | 0 |
| 0 | 6 | 66 |
| 0 | 17 | 83 |
G:=sub<GL(3,GF(89))| [1,0,0,0,69,54,0,0,40],[37,0,0,0,6,17,0,66,83] >;
C44⋊C8 in GAP, Magma, Sage, TeX
C_{44}\rtimes C_8 % in TeX
G:=Group("C44:C8"); // GroupNames label
G:=SmallGroup(352,10);
// by ID
G=gap.SmallGroup(352,10);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-11,24,121,55,86,11525]);
// Polycyclic
G:=Group<a,b|a^44=b^8=1,b*a*b^-1=a^-1>;
// generators/relations
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