metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C40:1C8, Dic5.7Q16, Dic5.12D8, C20.9M4(2), C8:1(C5:C8), C5:1(C8:1C8), C10.4(C4:C8), (C2xC8).10F5, (C2xC40).10C4, C20.41(C2xC8), C20:C8.8C2, C4.6(C4.F5), C2.1(D5.D8), C2.4(C20:C8), C10.5(C2.D8), (C2xDic5).28Q8, (C8xDic5).19C2, C22.16(C4:F5), C2.1(D10.Q8), C10.2(C8.C4), (C2xDic5).171D4, (C4xDic5).341C22, C4.7(C2xC5:C8), (C2xC5:2C8).20C4, (C2xC10).9(C4:C4), (C2xC4).118(C2xF5), (C2xC20).115(C2xC4), SmallGroup(320,220)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C40:1C8
G = < a,b | a40=b8=1, bab-1=a23 >
Quotients: C1, C2, C4, C22, C8, C2xC4, D4, Q8, C4:C4, C2xC8, M4(2), D8, Q16, F5, C4:C8, C2.D8, C8.C4, C5:C8, C2xF5, C8:1C8, C4.F5, C4:F5, C2xC5:C8, D5.D8, D10.Q8, C20:C8, C40:1C8
Smallest permutation representation of C40:1C8
►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)
(1 222 172 129 277 296 101 51)(2 229 181 152 278 303 110 74)(3 236 190 135 279 310 119 57)(4 203 199 158 280 317 88 80)(5 210 168 141 241 284 97 63)(6 217 177 124 242 291 106 46)(7 224 186 147 243 298 115 69)(8 231 195 130 244 305 84 52)(9 238 164 153 245 312 93 75)(10 205 173 136 246 319 102 58)(11 212 182 159 247 286 111 41)(12 219 191 142 248 293 120 64)(13 226 200 125 249 300 89 47)(14 233 169 148 250 307 98 70)(15 240 178 131 251 314 107 53)(16 207 187 154 252 281 116 76)(17 214 196 137 253 288 85 59)(18 221 165 160 254 295 94 42)(19 228 174 143 255 302 103 65)(20 235 183 126 256 309 112 48)(21 202 192 149 257 316 81 71)(22 209 161 132 258 283 90 54)(23 216 170 155 259 290 99 77)(24 223 179 138 260 297 108 60)(25 230 188 121 261 304 117 43)(26 237 197 144 262 311 86 66)(27 204 166 127 263 318 95 49)(28 211 175 150 264 285 104 72)(29 218 184 133 265 292 113 55)(30 225 193 156 266 299 82 78)(31 232 162 139 267 306 91 61)(32 239 171 122 268 313 100 44)(33 206 180 145 269 320 109 67)(34 213 189 128 270 287 118 50)(35 220 198 151 271 294 87 73)(36 227 167 134 272 301 96 56)(37 234 176 157 273 308 105 79)(38 201 185 140 274 315 114 62)(39 208 194 123 275 282 83 45)(40 215 163 146 276 289 92 68)
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), (1,222,172,129,277,296,101,51)(2,229,181,152,278,303,110,74)(3,236,190,135,279,310,119,57)(4,203,199,158,280,317,88,80)(5,210,168,141,241,284,97,63)(6,217,177,124,242,291,106,46)(7,224,186,147,243,298,115,69)(8,231,195,130,244,305,84,52)(9,238,164,153,245,312,93,75)(10,205,173,136,246,319,102,58)(11,212,182,159,247,286,111,41)(12,219,191,142,248,293,120,64)(13,226,200,125,249,300,89,47)(14,233,169,148,250,307,98,70)(15,240,178,131,251,314,107,53)(16,207,187,154,252,281,116,76)(17,214,196,137,253,288,85,59)(18,221,165,160,254,295,94,42)(19,228,174,143,255,302,103,65)(20,235,183,126,256,309,112,48)(21,202,192,149,257,316,81,71)(22,209,161,132,258,283,90,54)(23,216,170,155,259,290,99,77)(24,223,179,138,260,297,108,60)(25,230,188,121,261,304,117,43)(26,237,197,144,262,311,86,66)(27,204,166,127,263,318,95,49)(28,211,175,150,264,285,104,72)(29,218,184,133,265,292,113,55)(30,225,193,156,266,299,82,78)(31,232,162,139,267,306,91,61)(32,239,171,122,268,313,100,44)(33,206,180,145,269,320,109,67)(34,213,189,128,270,287,118,50)(35,220,198,151,271,294,87,73)(36,227,167,134,272,301,96,56)(37,234,176,157,273,308,105,79)(38,201,185,140,274,315,114,62)(39,208,194,123,275,282,83,45)(40,215,163,146,276,289,92,68)>;
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), (1,222,172,129,277,296,101,51)(2,229,181,152,278,303,110,74)(3,236,190,135,279,310,119,57)(4,203,199,158,280,317,88,80)(5,210,168,141,241,284,97,63)(6,217,177,124,242,291,106,46)(7,224,186,147,243,298,115,69)(8,231,195,130,244,305,84,52)(9,238,164,153,245,312,93,75)(10,205,173,136,246,319,102,58)(11,212,182,159,247,286,111,41)(12,219,191,142,248,293,120,64)(13,226,200,125,249,300,89,47)(14,233,169,148,250,307,98,70)(15,240,178,131,251,314,107,53)(16,207,187,154,252,281,116,76)(17,214,196,137,253,288,85,59)(18,221,165,160,254,295,94,42)(19,228,174,143,255,302,103,65)(20,235,183,126,256,309,112,48)(21,202,192,149,257,316,81,71)(22,209,161,132,258,283,90,54)(23,216,170,155,259,290,99,77)(24,223,179,138,260,297,108,60)(25,230,188,121,261,304,117,43)(26,237,197,144,262,311,86,66)(27,204,166,127,263,318,95,49)(28,211,175,150,264,285,104,72)(29,218,184,133,265,292,113,55)(30,225,193,156,266,299,82,78)(31,232,162,139,267,306,91,61)(32,239,171,122,268,313,100,44)(33,206,180,145,269,320,109,67)(34,213,189,128,270,287,118,50)(35,220,198,151,271,294,87,73)(36,227,167,134,272,301,96,56)(37,234,176,157,273,308,105,79)(38,201,185,140,274,315,114,62)(39,208,194,123,275,282,83,45)(40,215,163,146,276,289,92,68) );
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)], [(1,222,172,129,277,296,101,51),(2,229,181,152,278,303,110,74),(3,236,190,135,279,310,119,57),(4,203,199,158,280,317,88,80),(5,210,168,141,241,284,97,63),(6,217,177,124,242,291,106,46),(7,224,186,147,243,298,115,69),(8,231,195,130,244,305,84,52),(9,238,164,153,245,312,93,75),(10,205,173,136,246,319,102,58),(11,212,182,159,247,286,111,41),(12,219,191,142,248,293,120,64),(13,226,200,125,249,300,89,47),(14,233,169,148,250,307,98,70),(15,240,178,131,251,314,107,53),(16,207,187,154,252,281,116,76),(17,214,196,137,253,288,85,59),(18,221,165,160,254,295,94,42),(19,228,174,143,255,302,103,65),(20,235,183,126,256,309,112,48),(21,202,192,149,257,316,81,71),(22,209,161,132,258,283,90,54),(23,216,170,155,259,290,99,77),(24,223,179,138,260,297,108,60),(25,230,188,121,261,304,117,43),(26,237,197,144,262,311,86,66),(27,204,166,127,263,318,95,49),(28,211,175,150,264,285,104,72),(29,218,184,133,265,292,113,55),(30,225,193,156,266,299,82,78),(31,232,162,139,267,306,91,61),(32,239,171,122,268,313,100,44),(33,206,180,145,269,320,109,67),(34,213,189,128,270,287,118,50),(35,220,198,151,271,294,87,73),(36,227,167,134,272,301,96,56),(37,234,176,157,273,308,105,79),(38,201,185,140,274,315,114,62),(39,208,194,123,275,282,83,45),(40,215,163,146,276,289,92,68)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5 | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | ··· | 8P | 10A | 10B | 10C | 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 | 8 | 8 | 8 | ··· | 8 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 5 | 5 | 5 | 5 | 10 | 10 | 4 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | - | + | - | + | - | + | |||||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | D4 | Q8 | D8 | Q16 | M4(2) | C8.C4 | F5 | C5:C8 | C2xF5 | C4.F5 | C4:F5 | D5.D8 | D10.Q8 |
| kernel | C40:1C8 | C8xDic5 | C20:C8 | C2xC5:2C8 | C2xC40 | C40 | C2xDic5 | C2xDic5 | Dic5 | Dic5 | C20 | C10 | C2xC8 | C8 | C2xC4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 2 | 2 | 2 | 8 | 1 | 1 | 2 | 2 | 2 | 4 | 1 | 2 | 1 | 2 | 2 | 4 | 4 |
Matrix representation of C40:1C8 ►in GL6(F41)
| 40 | 2 | 0 | 0 | 0 | 0 |
| 40 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 37 | 27 | 37 |
| 0 | 0 | 4 | 4 | 0 | 31 |
| 0 | 0 | 10 | 14 | 14 | 10 |
| 0 | 0 | 31 | 0 | 4 | 4 |
| 0 | 10 | 0 | 0 | 0 | 0 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 26 | 6 | 25 |
| 0 | 0 | 21 | 40 | 15 | 9 |
| 0 | 0 | 16 | 10 | 1 | 22 |
| 0 | 0 | 32 | 12 | 31 | 6 |
G:=sub<GL(6,GF(41))| [40,40,0,0,0,0,2,1,0,0,0,0,0,0,0,4,10,31,0,0,37,4,14,0,0,0,27,0,14,4,0,0,37,31,10,4],[0,5,0,0,0,0,10,0,0,0,0,0,0,0,35,21,16,32,0,0,26,40,10,12,0,0,6,15,1,31,0,0,25,9,22,6] >;
C40:1C8 in GAP, Magma, Sage, TeX
C_{40}\rtimes_1C_8 % in TeX
G:=Group("C40:1C8"); // GroupNames label
G:=SmallGroup(320,220);
// by ID
G=gap.SmallGroup(320,220);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,288,100,1123,136,6278,3156]);
// Polycyclic
G:=Group<a,b|a^40=b^8=1,b*a*b^-1=a^23>;
// generators/relations
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