metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C41⋊3C8, C82.2C4, C4.2D41, C2.Dic41, C164.2C2, SmallGroup(328,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C41 — C41⋊3C8 |
Generators and relations for C41⋊3C8
G = < a,b | a41=b8=1, bab-1=a-1 >
Smallest permutation representation of C41⋊3C8
►Regular action on 328 points
Generators in S328
(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)
(1 288 126 206 47 247 116 165)(2 328 127 246 48 287 117 205)(3 327 128 245 49 286 118 204)(4 326 129 244 50 285 119 203)(5 325 130 243 51 284 120 202)(6 324 131 242 52 283 121 201)(7 323 132 241 53 282 122 200)(8 322 133 240 54 281 123 199)(9 321 134 239 55 280 83 198)(10 320 135 238 56 279 84 197)(11 319 136 237 57 278 85 196)(12 318 137 236 58 277 86 195)(13 317 138 235 59 276 87 194)(14 316 139 234 60 275 88 193)(15 315 140 233 61 274 89 192)(16 314 141 232 62 273 90 191)(17 313 142 231 63 272 91 190)(18 312 143 230 64 271 92 189)(19 311 144 229 65 270 93 188)(20 310 145 228 66 269 94 187)(21 309 146 227 67 268 95 186)(22 308 147 226 68 267 96 185)(23 307 148 225 69 266 97 184)(24 306 149 224 70 265 98 183)(25 305 150 223 71 264 99 182)(26 304 151 222 72 263 100 181)(27 303 152 221 73 262 101 180)(28 302 153 220 74 261 102 179)(29 301 154 219 75 260 103 178)(30 300 155 218 76 259 104 177)(31 299 156 217 77 258 105 176)(32 298 157 216 78 257 106 175)(33 297 158 215 79 256 107 174)(34 296 159 214 80 255 108 173)(35 295 160 213 81 254 109 172)(36 294 161 212 82 253 110 171)(37 293 162 211 42 252 111 170)(38 292 163 210 43 251 112 169)(39 291 164 209 44 250 113 168)(40 290 124 208 45 249 114 167)(41 289 125 207 46 248 115 166)
G:=sub<Sym(328)| (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), (1,288,126,206,47,247,116,165)(2,328,127,246,48,287,117,205)(3,327,128,245,49,286,118,204)(4,326,129,244,50,285,119,203)(5,325,130,243,51,284,120,202)(6,324,131,242,52,283,121,201)(7,323,132,241,53,282,122,200)(8,322,133,240,54,281,123,199)(9,321,134,239,55,280,83,198)(10,320,135,238,56,279,84,197)(11,319,136,237,57,278,85,196)(12,318,137,236,58,277,86,195)(13,317,138,235,59,276,87,194)(14,316,139,234,60,275,88,193)(15,315,140,233,61,274,89,192)(16,314,141,232,62,273,90,191)(17,313,142,231,63,272,91,190)(18,312,143,230,64,271,92,189)(19,311,144,229,65,270,93,188)(20,310,145,228,66,269,94,187)(21,309,146,227,67,268,95,186)(22,308,147,226,68,267,96,185)(23,307,148,225,69,266,97,184)(24,306,149,224,70,265,98,183)(25,305,150,223,71,264,99,182)(26,304,151,222,72,263,100,181)(27,303,152,221,73,262,101,180)(28,302,153,220,74,261,102,179)(29,301,154,219,75,260,103,178)(30,300,155,218,76,259,104,177)(31,299,156,217,77,258,105,176)(32,298,157,216,78,257,106,175)(33,297,158,215,79,256,107,174)(34,296,159,214,80,255,108,173)(35,295,160,213,81,254,109,172)(36,294,161,212,82,253,110,171)(37,293,162,211,42,252,111,170)(38,292,163,210,43,251,112,169)(39,291,164,209,44,250,113,168)(40,290,124,208,45,249,114,167)(41,289,125,207,46,248,115,166)>;
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), (1,288,126,206,47,247,116,165)(2,328,127,246,48,287,117,205)(3,327,128,245,49,286,118,204)(4,326,129,244,50,285,119,203)(5,325,130,243,51,284,120,202)(6,324,131,242,52,283,121,201)(7,323,132,241,53,282,122,200)(8,322,133,240,54,281,123,199)(9,321,134,239,55,280,83,198)(10,320,135,238,56,279,84,197)(11,319,136,237,57,278,85,196)(12,318,137,236,58,277,86,195)(13,317,138,235,59,276,87,194)(14,316,139,234,60,275,88,193)(15,315,140,233,61,274,89,192)(16,314,141,232,62,273,90,191)(17,313,142,231,63,272,91,190)(18,312,143,230,64,271,92,189)(19,311,144,229,65,270,93,188)(20,310,145,228,66,269,94,187)(21,309,146,227,67,268,95,186)(22,308,147,226,68,267,96,185)(23,307,148,225,69,266,97,184)(24,306,149,224,70,265,98,183)(25,305,150,223,71,264,99,182)(26,304,151,222,72,263,100,181)(27,303,152,221,73,262,101,180)(28,302,153,220,74,261,102,179)(29,301,154,219,75,260,103,178)(30,300,155,218,76,259,104,177)(31,299,156,217,77,258,105,176)(32,298,157,216,78,257,106,175)(33,297,158,215,79,256,107,174)(34,296,159,214,80,255,108,173)(35,295,160,213,81,254,109,172)(36,294,161,212,82,253,110,171)(37,293,162,211,42,252,111,170)(38,292,163,210,43,251,112,169)(39,291,164,209,44,250,113,168)(40,290,124,208,45,249,114,167)(41,289,125,207,46,248,115,166) );
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)], [(1,288,126,206,47,247,116,165),(2,328,127,246,48,287,117,205),(3,327,128,245,49,286,118,204),(4,326,129,244,50,285,119,203),(5,325,130,243,51,284,120,202),(6,324,131,242,52,283,121,201),(7,323,132,241,53,282,122,200),(8,322,133,240,54,281,123,199),(9,321,134,239,55,280,83,198),(10,320,135,238,56,279,84,197),(11,319,136,237,57,278,85,196),(12,318,137,236,58,277,86,195),(13,317,138,235,59,276,87,194),(14,316,139,234,60,275,88,193),(15,315,140,233,61,274,89,192),(16,314,141,232,62,273,90,191),(17,313,142,231,63,272,91,190),(18,312,143,230,64,271,92,189),(19,311,144,229,65,270,93,188),(20,310,145,228,66,269,94,187),(21,309,146,227,67,268,95,186),(22,308,147,226,68,267,96,185),(23,307,148,225,69,266,97,184),(24,306,149,224,70,265,98,183),(25,305,150,223,71,264,99,182),(26,304,151,222,72,263,100,181),(27,303,152,221,73,262,101,180),(28,302,153,220,74,261,102,179),(29,301,154,219,75,260,103,178),(30,300,155,218,76,259,104,177),(31,299,156,217,77,258,105,176),(32,298,157,216,78,257,106,175),(33,297,158,215,79,256,107,174),(34,296,159,214,80,255,108,173),(35,295,160,213,81,254,109,172),(36,294,161,212,82,253,110,171),(37,293,162,211,42,252,111,170),(38,292,163,210,43,251,112,169),(39,291,164,209,44,250,113,168),(40,290,124,208,45,249,114,167),(41,289,125,207,46,248,115,166)]])
88 conjugacy classes
| class | 1 | 2 | 4A | 4B | 8A | 8B | 8C | 8D | 41A | ··· | 41T | 82A | ··· | 82T | 164A | ··· | 164AN |
| order | 1 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 41 | ··· | 41 | 82 | ··· | 82 | 164 | ··· | 164 |
| size | 1 | 1 | 1 | 1 | 41 | 41 | 41 | 41 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
88 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | - | |||
| image | C1 | C2 | C4 | C8 | D41 | Dic41 | C41⋊3C8 |
| kernel | C41⋊3C8 | C164 | C82 | C41 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 20 | 20 | 40 |
Matrix representation of C41⋊3C8 ►in GL2(𝔽2297) generated by
| 0 | 1 |
| 2296 | 316 |
| 1581 | 449 |
| 1596 | 716 |
G:=sub<GL(2,GF(2297))| [0,2296,1,316],[1581,1596,449,716] >;
C41⋊3C8 in GAP, Magma, Sage, TeX
C_{41}\rtimes_3C_8 % in TeX
G:=Group("C41:3C8"); // GroupNames label
G:=SmallGroup(328,1);
// by ID
G=gap.SmallGroup(328,1);
# by ID
G:=PCGroup([4,-2,-2,-2,-41,8,21,5123]);
// Polycyclic
G:=Group<a,b|a^41=b^8=1,b*a*b^-1=a^-1>;
// generators/relations
Export