metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C40.6C8, C20.22C42, C10.1M5(2), C5⋊C16⋊7C4, C8.3(C5⋊C8), C5⋊3(C16⋊5C4), C4.18(C4×F5), (C2×C8).18F5, C20.48(C2×C8), C10.10(C4×C8), (C2×C40).21C4, (C2×Dic5).9C8, C22.8(D5⋊C8), (C4×Dic5).41C4, (C8×Dic5).26C2, C2.1(C8.F5), C2.6(C4×C5⋊C8), C4.15(C2×C5⋊C8), (C2×C5⋊C16).3C2, (C2×C10).4(C2×C8), C5⋊2C8.37(C2×C4), (C2×C4).154(C2×F5), (C2×C20).160(C2×C4), (C2×C5⋊2C8).344C22, SmallGroup(320,224)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C40.C8
G = < a,b | a40=1, b8=a20, bab-1=a13 >
Smallest permutation representation of C40.C8
►Regular action on 320 points
Generators in S320
(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)
(1 196 66 295 210 100 253 154 21 176 46 315 230 120 273 134)(2 193 75 308 211 97 262 127 22 173 55 288 231 117 242 147)(3 190 44 281 212 94 271 140 23 170 64 301 232 114 251 160)(4 187 53 294 213 91 280 153 24 167 73 314 233 111 260 133)(5 184 62 307 214 88 249 126 25 164 42 287 234 108 269 146)(6 181 71 320 215 85 258 139 26 161 51 300 235 105 278 159)(7 178 80 293 216 82 267 152 27 198 60 313 236 102 247 132)(8 175 49 306 217 119 276 125 28 195 69 286 237 99 256 145)(9 172 58 319 218 116 245 138 29 192 78 299 238 96 265 158)(10 169 67 292 219 113 254 151 30 189 47 312 239 93 274 131)(11 166 76 305 220 110 263 124 31 186 56 285 240 90 243 144)(12 163 45 318 221 107 272 137 32 183 65 298 201 87 252 157)(13 200 54 291 222 104 241 150 33 180 74 311 202 84 261 130)(14 197 63 304 223 101 250 123 34 177 43 284 203 81 270 143)(15 194 72 317 224 98 259 136 35 174 52 297 204 118 279 156)(16 191 41 290 225 95 268 149 36 171 61 310 205 115 248 129)(17 188 50 303 226 92 277 122 37 168 70 283 206 112 257 142)(18 185 59 316 227 89 246 135 38 165 79 296 207 109 266 155)(19 182 68 289 228 86 255 148 39 162 48 309 208 106 275 128)(20 179 77 302 229 83 264 121 40 199 57 282 209 103 244 141)
G:=sub<Sym(320)| (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), (1,196,66,295,210,100,253,154,21,176,46,315,230,120,273,134)(2,193,75,308,211,97,262,127,22,173,55,288,231,117,242,147)(3,190,44,281,212,94,271,140,23,170,64,301,232,114,251,160)(4,187,53,294,213,91,280,153,24,167,73,314,233,111,260,133)(5,184,62,307,214,88,249,126,25,164,42,287,234,108,269,146)(6,181,71,320,215,85,258,139,26,161,51,300,235,105,278,159)(7,178,80,293,216,82,267,152,27,198,60,313,236,102,247,132)(8,175,49,306,217,119,276,125,28,195,69,286,237,99,256,145)(9,172,58,319,218,116,245,138,29,192,78,299,238,96,265,158)(10,169,67,292,219,113,254,151,30,189,47,312,239,93,274,131)(11,166,76,305,220,110,263,124,31,186,56,285,240,90,243,144)(12,163,45,318,221,107,272,137,32,183,65,298,201,87,252,157)(13,200,54,291,222,104,241,150,33,180,74,311,202,84,261,130)(14,197,63,304,223,101,250,123,34,177,43,284,203,81,270,143)(15,194,72,317,224,98,259,136,35,174,52,297,204,118,279,156)(16,191,41,290,225,95,268,149,36,171,61,310,205,115,248,129)(17,188,50,303,226,92,277,122,37,168,70,283,206,112,257,142)(18,185,59,316,227,89,246,135,38,165,79,296,207,109,266,155)(19,182,68,289,228,86,255,148,39,162,48,309,208,106,275,128)(20,179,77,302,229,83,264,121,40,199,57,282,209,103,244,141)>;
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), (1,196,66,295,210,100,253,154,21,176,46,315,230,120,273,134)(2,193,75,308,211,97,262,127,22,173,55,288,231,117,242,147)(3,190,44,281,212,94,271,140,23,170,64,301,232,114,251,160)(4,187,53,294,213,91,280,153,24,167,73,314,233,111,260,133)(5,184,62,307,214,88,249,126,25,164,42,287,234,108,269,146)(6,181,71,320,215,85,258,139,26,161,51,300,235,105,278,159)(7,178,80,293,216,82,267,152,27,198,60,313,236,102,247,132)(8,175,49,306,217,119,276,125,28,195,69,286,237,99,256,145)(9,172,58,319,218,116,245,138,29,192,78,299,238,96,265,158)(10,169,67,292,219,113,254,151,30,189,47,312,239,93,274,131)(11,166,76,305,220,110,263,124,31,186,56,285,240,90,243,144)(12,163,45,318,221,107,272,137,32,183,65,298,201,87,252,157)(13,200,54,291,222,104,241,150,33,180,74,311,202,84,261,130)(14,197,63,304,223,101,250,123,34,177,43,284,203,81,270,143)(15,194,72,317,224,98,259,136,35,174,52,297,204,118,279,156)(16,191,41,290,225,95,268,149,36,171,61,310,205,115,248,129)(17,188,50,303,226,92,277,122,37,168,70,283,206,112,257,142)(18,185,59,316,227,89,246,135,38,165,79,296,207,109,266,155)(19,182,68,289,228,86,255,148,39,162,48,309,208,106,275,128)(20,179,77,302,229,83,264,121,40,199,57,282,209,103,244,141) );
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)], [(1,196,66,295,210,100,253,154,21,176,46,315,230,120,273,134),(2,193,75,308,211,97,262,127,22,173,55,288,231,117,242,147),(3,190,44,281,212,94,271,140,23,170,64,301,232,114,251,160),(4,187,53,294,213,91,280,153,24,167,73,314,233,111,260,133),(5,184,62,307,214,88,249,126,25,164,42,287,234,108,269,146),(6,181,71,320,215,85,258,139,26,161,51,300,235,105,278,159),(7,178,80,293,216,82,267,152,27,198,60,313,236,102,247,132),(8,175,49,306,217,119,276,125,28,195,69,286,237,99,256,145),(9,172,58,319,218,116,245,138,29,192,78,299,238,96,265,158),(10,169,67,292,219,113,254,151,30,189,47,312,239,93,274,131),(11,166,76,305,220,110,263,124,31,186,56,285,240,90,243,144),(12,163,45,318,221,107,272,137,32,183,65,298,201,87,252,157),(13,200,54,291,222,104,241,150,33,180,74,311,202,84,261,130),(14,197,63,304,223,101,250,123,34,177,43,284,203,81,270,143),(15,194,72,317,224,98,259,136,35,174,52,297,204,118,279,156),(16,191,41,290,225,95,268,149,36,171,61,310,205,115,248,129),(17,188,50,303,226,92,277,122,37,168,70,283,206,112,257,142),(18,185,59,316,227,89,246,135,38,165,79,296,207,109,266,155),(19,182,68,289,228,86,255,148,39,162,48,309,208,106,275,128),(20,179,77,302,229,83,264,121,40,199,57,282,209,103,244,141)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5 | 8A | 8B | 8C | 8D | 8E | ··· | 8L | 10A | 10B | 10C | 16A | ··· | 16P | 20A | 20B | 20C | 20D | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 10 | 10 | 10 | 16 | ··· | 16 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 10 | 10 | 10 | 10 | 4 | 2 | 2 | 2 | 2 | 5 | ··· | 5 | 4 | 4 | 4 | 10 | ··· | 10 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | - | + | |||||||||
| image | C1 | C2 | C2 | C4 | C4 | C4 | C8 | C8 | M5(2) | F5 | C5⋊C8 | C2×F5 | C4×F5 | D5⋊C8 | C8.F5 |
| kernel | C40.C8 | C8×Dic5 | C2×C5⋊C16 | C5⋊C16 | C4×Dic5 | C2×C40 | C40 | C2×Dic5 | C10 | C2×C8 | C8 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 1 | 2 | 8 | 2 | 2 | 8 | 8 | 8 | 1 | 2 | 1 | 2 | 2 | 8 |
Matrix representation of C40.C8 ►in GL6(𝔽241)
| 29 | 113 | 0 | 0 | 0 | 0 |
| 201 | 212 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 240 | 240 | 240 | 240 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 150 | 47 | 0 | 0 | 0 | 0 |
| 106 | 91 | 0 | 0 | 0 | 0 |
| 0 | 0 | 133 | 12 | 46 | 185 |
| 0 | 0 | 34 | 173 | 229 | 121 |
| 0 | 0 | 56 | 189 | 68 | 102 |
| 0 | 0 | 120 | 154 | 52 | 108 |
G:=sub<GL(6,GF(241))| [29,201,0,0,0,0,113,212,0,0,0,0,0,0,0,0,240,1,0,0,0,0,240,0,0,0,1,0,240,0,0,0,0,1,240,0],[150,106,0,0,0,0,47,91,0,0,0,0,0,0,133,34,56,120,0,0,12,173,189,154,0,0,46,229,68,52,0,0,185,121,102,108] >;
C40.C8 in GAP, Magma, Sage, TeX
C_{40}.C_8 % in TeX
G:=Group("C40.C8"); // GroupNames label
G:=SmallGroup(320,224);
// by ID
G=gap.SmallGroup(320,224);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,477,176,100,102,6278,3156]);
// Polycyclic
G:=Group<a,b|a^40=1,b^8=a^20,b*a*b^-1=a^13>;
// generators/relations
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