metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C40.10C8, C20.44C42, C42.4Dic5, C10.10M5(2), C5⋊2C16⋊11C4, C8.39(C4×D5), (C4×C8).15D5, C5⋊4(C16⋊5C4), C8.2(C5⋊2C8), (C4×C20).35C4, C40.97(C2×C4), (C2×C40).47C4, (C2×C20).12C8, C10.15(C4×C8), (C4×C40).19C2, C20.75(C2×C8), (C2×C8).330D10, (C2×C8).13Dic5, C4.14(C4×Dic5), C2.1(C20.4C8), (C2×C40).395C22, C2.4(C4×C5⋊2C8), C4.13(C2×C5⋊2C8), (C2×C5⋊2C16).7C2, (C2×C10).54(C2×C8), (C2×C4).2(C5⋊2C8), C22.8(C2×C5⋊2C8), (C2×C20).483(C2×C4), (C2×C4).91(C2×Dic5), SmallGroup(320,19)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C40.10C8
G = < a,b | a40=1, b8=a20, bab-1=a29 >
Smallest permutation representation of C40.10C8
►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)
(1 235 159 195 94 254 57 299 21 215 139 175 114 274 77 319)(2 224 160 184 95 243 58 288 22 204 140 164 115 263 78 308)(3 213 121 173 96 272 59 317 23 233 141 193 116 252 79 297)(4 202 122 162 97 261 60 306 24 222 142 182 117 241 80 286)(5 231 123 191 98 250 61 295 25 211 143 171 118 270 41 315)(6 220 124 180 99 279 62 284 26 240 144 200 119 259 42 304)(7 209 125 169 100 268 63 313 27 229 145 189 120 248 43 293)(8 238 126 198 101 257 64 302 28 218 146 178 81 277 44 282)(9 227 127 187 102 246 65 291 29 207 147 167 82 266 45 311)(10 216 128 176 103 275 66 320 30 236 148 196 83 255 46 300)(11 205 129 165 104 264 67 309 31 225 149 185 84 244 47 289)(12 234 130 194 105 253 68 298 32 214 150 174 85 273 48 318)(13 223 131 183 106 242 69 287 33 203 151 163 86 262 49 307)(14 212 132 172 107 271 70 316 34 232 152 192 87 251 50 296)(15 201 133 161 108 260 71 305 35 221 153 181 88 280 51 285)(16 230 134 190 109 249 72 294 36 210 154 170 89 269 52 314)(17 219 135 179 110 278 73 283 37 239 155 199 90 258 53 303)(18 208 136 168 111 267 74 312 38 228 156 188 91 247 54 292)(19 237 137 197 112 256 75 301 39 217 157 177 92 276 55 281)(20 226 138 186 113 245 76 290 40 206 158 166 93 265 56 310)
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), (1,235,159,195,94,254,57,299,21,215,139,175,114,274,77,319)(2,224,160,184,95,243,58,288,22,204,140,164,115,263,78,308)(3,213,121,173,96,272,59,317,23,233,141,193,116,252,79,297)(4,202,122,162,97,261,60,306,24,222,142,182,117,241,80,286)(5,231,123,191,98,250,61,295,25,211,143,171,118,270,41,315)(6,220,124,180,99,279,62,284,26,240,144,200,119,259,42,304)(7,209,125,169,100,268,63,313,27,229,145,189,120,248,43,293)(8,238,126,198,101,257,64,302,28,218,146,178,81,277,44,282)(9,227,127,187,102,246,65,291,29,207,147,167,82,266,45,311)(10,216,128,176,103,275,66,320,30,236,148,196,83,255,46,300)(11,205,129,165,104,264,67,309,31,225,149,185,84,244,47,289)(12,234,130,194,105,253,68,298,32,214,150,174,85,273,48,318)(13,223,131,183,106,242,69,287,33,203,151,163,86,262,49,307)(14,212,132,172,107,271,70,316,34,232,152,192,87,251,50,296)(15,201,133,161,108,260,71,305,35,221,153,181,88,280,51,285)(16,230,134,190,109,249,72,294,36,210,154,170,89,269,52,314)(17,219,135,179,110,278,73,283,37,239,155,199,90,258,53,303)(18,208,136,168,111,267,74,312,38,228,156,188,91,247,54,292)(19,237,137,197,112,256,75,301,39,217,157,177,92,276,55,281)(20,226,138,186,113,245,76,290,40,206,158,166,93,265,56,310)>;
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), (1,235,159,195,94,254,57,299,21,215,139,175,114,274,77,319)(2,224,160,184,95,243,58,288,22,204,140,164,115,263,78,308)(3,213,121,173,96,272,59,317,23,233,141,193,116,252,79,297)(4,202,122,162,97,261,60,306,24,222,142,182,117,241,80,286)(5,231,123,191,98,250,61,295,25,211,143,171,118,270,41,315)(6,220,124,180,99,279,62,284,26,240,144,200,119,259,42,304)(7,209,125,169,100,268,63,313,27,229,145,189,120,248,43,293)(8,238,126,198,101,257,64,302,28,218,146,178,81,277,44,282)(9,227,127,187,102,246,65,291,29,207,147,167,82,266,45,311)(10,216,128,176,103,275,66,320,30,236,148,196,83,255,46,300)(11,205,129,165,104,264,67,309,31,225,149,185,84,244,47,289)(12,234,130,194,105,253,68,298,32,214,150,174,85,273,48,318)(13,223,131,183,106,242,69,287,33,203,151,163,86,262,49,307)(14,212,132,172,107,271,70,316,34,232,152,192,87,251,50,296)(15,201,133,161,108,260,71,305,35,221,153,181,88,280,51,285)(16,230,134,190,109,249,72,294,36,210,154,170,89,269,52,314)(17,219,135,179,110,278,73,283,37,239,155,199,90,258,53,303)(18,208,136,168,111,267,74,312,38,228,156,188,91,247,54,292)(19,237,137,197,112,256,75,301,39,217,157,177,92,276,55,281)(20,226,138,186,113,245,76,290,40,206,158,166,93,265,56,310) );
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)], [(1,235,159,195,94,254,57,299,21,215,139,175,114,274,77,319),(2,224,160,184,95,243,58,288,22,204,140,164,115,263,78,308),(3,213,121,173,96,272,59,317,23,233,141,193,116,252,79,297),(4,202,122,162,97,261,60,306,24,222,142,182,117,241,80,286),(5,231,123,191,98,250,61,295,25,211,143,171,118,270,41,315),(6,220,124,180,99,279,62,284,26,240,144,200,119,259,42,304),(7,209,125,169,100,268,63,313,27,229,145,189,120,248,43,293),(8,238,126,198,101,257,64,302,28,218,146,178,81,277,44,282),(9,227,127,187,102,246,65,291,29,207,147,167,82,266,45,311),(10,216,128,176,103,275,66,320,30,236,148,196,83,255,46,300),(11,205,129,165,104,264,67,309,31,225,149,185,84,244,47,289),(12,234,130,194,105,253,68,298,32,214,150,174,85,273,48,318),(13,223,131,183,106,242,69,287,33,203,151,163,86,262,49,307),(14,212,132,172,107,271,70,316,34,232,152,192,87,251,50,296),(15,201,133,161,108,260,71,305,35,221,153,181,88,280,51,285),(16,230,134,190,109,249,72,294,36,210,154,170,89,269,52,314),(17,219,135,179,110,278,73,283,37,239,155,199,90,258,53,303),(18,208,136,168,111,267,74,312,38,228,156,188,91,247,54,292),(19,237,137,197,112,256,75,301,39,217,157,177,92,276,55,281),(20,226,138,186,113,245,76,290,40,206,158,166,93,265,56,310)]])
104 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 10A | ··· | 10F | 16A | ··· | 16P | 20A | ··· | 20X | 40A | ··· | 40AF |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 16 | ··· | 16 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 10 | ··· | 10 | 2 | ··· | 2 | 2 | ··· | 2 |
104 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | - | + | ||||||||||
| image | C1 | C2 | C2 | C4 | C4 | C4 | C8 | C8 | D5 | Dic5 | Dic5 | D10 | M5(2) | C5⋊2C8 | C4×D5 | C5⋊2C8 | C20.4C8 |
| kernel | C40.10C8 | C2×C5⋊2C16 | C4×C40 | C5⋊2C16 | C4×C20 | C2×C40 | C40 | C2×C20 | C4×C8 | C42 | C2×C8 | C2×C8 | C10 | C8 | C8 | C2×C4 | C2 |
| # reps | 1 | 2 | 1 | 8 | 2 | 2 | 8 | 8 | 2 | 2 | 2 | 2 | 8 | 8 | 8 | 8 | 32 |
Matrix representation of C40.10C8 ►in GL4(𝔽241) generated by
| 0 | 30 | 0 | 0 |
| 211 | 0 | 0 | 0 |
| 0 | 0 | 190 | 52 |
| 0 | 0 | 190 | 0 |
| 71 | 131 | 0 | 0 |
| 131 | 170 | 0 | 0 |
| 0 | 0 | 93 | 157 |
| 0 | 0 | 32 | 148 |
G:=sub<GL(4,GF(241))| [0,211,0,0,30,0,0,0,0,0,190,190,0,0,52,0],[71,131,0,0,131,170,0,0,0,0,93,32,0,0,157,148] >;
C40.10C8 in GAP, Magma, Sage, TeX
C_{40}._{10}C_8 % in TeX
G:=Group("C40.10C8"); // GroupNames label
G:=SmallGroup(320,19);
// by ID
G=gap.SmallGroup(320,19);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,477,64,100,102,12550]);
// Polycyclic
G:=Group<a,b|a^40=1,b^8=a^20,b*a*b^-1=a^29>;
// generators/relations
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