metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C80⋊17C4, C16⋊5Dic5, C20.47C42, C10.8M5(2), C5⋊2C8.3C8, C5⋊2C16⋊12C4, C4.21(C8×D5), C8.41(C4×D5), (C2×C16).8D5, C5⋊5(C16⋊5C4), C10.19(C4×C8), C20.60(C2×C8), (C2×C80).15C2, C40.99(C2×C4), C2.4(C8×Dic5), (C2×C8).334D10, (C8×Dic5).9C2, (C2×Dic5).4C8, C4.16(C4×Dic5), C8.24(C2×Dic5), C2.2(C80⋊C2), C22.10(C8×D5), (C4×Dic5).14C4, (C2×C40).400C22, (C2×C5⋊2C8).16C4, (C2×C10).39(C2×C8), (C2×C4).168(C4×D5), (C2×C5⋊2C16).10C2, (C2×C20).415(C2×C4), SmallGroup(320,60)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C80⋊17C4
G = < a,b | a80=b4=1, bab-1=a9 >
Smallest permutation representation of C80⋊17C4
►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)
(1 266 176 142)(2 275 177 151)(3 284 178 160)(4 293 179 89)(5 302 180 98)(6 311 181 107)(7 320 182 116)(8 249 183 125)(9 258 184 134)(10 267 185 143)(11 276 186 152)(12 285 187 81)(13 294 188 90)(14 303 189 99)(15 312 190 108)(16 241 191 117)(17 250 192 126)(18 259 193 135)(19 268 194 144)(20 277 195 153)(21 286 196 82)(22 295 197 91)(23 304 198 100)(24 313 199 109)(25 242 200 118)(26 251 201 127)(27 260 202 136)(28 269 203 145)(29 278 204 154)(30 287 205 83)(31 296 206 92)(32 305 207 101)(33 314 208 110)(34 243 209 119)(35 252 210 128)(36 261 211 137)(37 270 212 146)(38 279 213 155)(39 288 214 84)(40 297 215 93)(41 306 216 102)(42 315 217 111)(43 244 218 120)(44 253 219 129)(45 262 220 138)(46 271 221 147)(47 280 222 156)(48 289 223 85)(49 298 224 94)(50 307 225 103)(51 316 226 112)(52 245 227 121)(53 254 228 130)(54 263 229 139)(55 272 230 148)(56 281 231 157)(57 290 232 86)(58 299 233 95)(59 308 234 104)(60 317 235 113)(61 246 236 122)(62 255 237 131)(63 264 238 140)(64 273 239 149)(65 282 240 158)(66 291 161 87)(67 300 162 96)(68 309 163 105)(69 318 164 114)(70 247 165 123)(71 256 166 132)(72 265 167 141)(73 274 168 150)(74 283 169 159)(75 292 170 88)(76 301 171 97)(77 310 172 106)(78 319 173 115)(79 248 174 124)(80 257 175 133)
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), (1,266,176,142)(2,275,177,151)(3,284,178,160)(4,293,179,89)(5,302,180,98)(6,311,181,107)(7,320,182,116)(8,249,183,125)(9,258,184,134)(10,267,185,143)(11,276,186,152)(12,285,187,81)(13,294,188,90)(14,303,189,99)(15,312,190,108)(16,241,191,117)(17,250,192,126)(18,259,193,135)(19,268,194,144)(20,277,195,153)(21,286,196,82)(22,295,197,91)(23,304,198,100)(24,313,199,109)(25,242,200,118)(26,251,201,127)(27,260,202,136)(28,269,203,145)(29,278,204,154)(30,287,205,83)(31,296,206,92)(32,305,207,101)(33,314,208,110)(34,243,209,119)(35,252,210,128)(36,261,211,137)(37,270,212,146)(38,279,213,155)(39,288,214,84)(40,297,215,93)(41,306,216,102)(42,315,217,111)(43,244,218,120)(44,253,219,129)(45,262,220,138)(46,271,221,147)(47,280,222,156)(48,289,223,85)(49,298,224,94)(50,307,225,103)(51,316,226,112)(52,245,227,121)(53,254,228,130)(54,263,229,139)(55,272,230,148)(56,281,231,157)(57,290,232,86)(58,299,233,95)(59,308,234,104)(60,317,235,113)(61,246,236,122)(62,255,237,131)(63,264,238,140)(64,273,239,149)(65,282,240,158)(66,291,161,87)(67,300,162,96)(68,309,163,105)(69,318,164,114)(70,247,165,123)(71,256,166,132)(72,265,167,141)(73,274,168,150)(74,283,169,159)(75,292,170,88)(76,301,171,97)(77,310,172,106)(78,319,173,115)(79,248,174,124)(80,257,175,133)>;
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), (1,266,176,142)(2,275,177,151)(3,284,178,160)(4,293,179,89)(5,302,180,98)(6,311,181,107)(7,320,182,116)(8,249,183,125)(9,258,184,134)(10,267,185,143)(11,276,186,152)(12,285,187,81)(13,294,188,90)(14,303,189,99)(15,312,190,108)(16,241,191,117)(17,250,192,126)(18,259,193,135)(19,268,194,144)(20,277,195,153)(21,286,196,82)(22,295,197,91)(23,304,198,100)(24,313,199,109)(25,242,200,118)(26,251,201,127)(27,260,202,136)(28,269,203,145)(29,278,204,154)(30,287,205,83)(31,296,206,92)(32,305,207,101)(33,314,208,110)(34,243,209,119)(35,252,210,128)(36,261,211,137)(37,270,212,146)(38,279,213,155)(39,288,214,84)(40,297,215,93)(41,306,216,102)(42,315,217,111)(43,244,218,120)(44,253,219,129)(45,262,220,138)(46,271,221,147)(47,280,222,156)(48,289,223,85)(49,298,224,94)(50,307,225,103)(51,316,226,112)(52,245,227,121)(53,254,228,130)(54,263,229,139)(55,272,230,148)(56,281,231,157)(57,290,232,86)(58,299,233,95)(59,308,234,104)(60,317,235,113)(61,246,236,122)(62,255,237,131)(63,264,238,140)(64,273,239,149)(65,282,240,158)(66,291,161,87)(67,300,162,96)(68,309,163,105)(69,318,164,114)(70,247,165,123)(71,256,166,132)(72,265,167,141)(73,274,168,150)(74,283,169,159)(75,292,170,88)(76,301,171,97)(77,310,172,106)(78,319,173,115)(79,248,174,124)(80,257,175,133) );
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)], [(1,266,176,142),(2,275,177,151),(3,284,178,160),(4,293,179,89),(5,302,180,98),(6,311,181,107),(7,320,182,116),(8,249,183,125),(9,258,184,134),(10,267,185,143),(11,276,186,152),(12,285,187,81),(13,294,188,90),(14,303,189,99),(15,312,190,108),(16,241,191,117),(17,250,192,126),(18,259,193,135),(19,268,194,144),(20,277,195,153),(21,286,196,82),(22,295,197,91),(23,304,198,100),(24,313,199,109),(25,242,200,118),(26,251,201,127),(27,260,202,136),(28,269,203,145),(29,278,204,154),(30,287,205,83),(31,296,206,92),(32,305,207,101),(33,314,208,110),(34,243,209,119),(35,252,210,128),(36,261,211,137),(37,270,212,146),(38,279,213,155),(39,288,214,84),(40,297,215,93),(41,306,216,102),(42,315,217,111),(43,244,218,120),(44,253,219,129),(45,262,220,138),(46,271,221,147),(47,280,222,156),(48,289,223,85),(49,298,224,94),(50,307,225,103),(51,316,226,112),(52,245,227,121),(53,254,228,130),(54,263,229,139),(55,272,230,148),(56,281,231,157),(57,290,232,86),(58,299,233,95),(59,308,234,104),(60,317,235,113),(61,246,236,122),(62,255,237,131),(63,264,238,140),(64,273,239,149),(65,282,240,158),(66,291,161,87),(67,300,162,96),(68,309,163,105),(69,318,164,114),(70,247,165,123),(71,256,166,132),(72,265,167,141),(73,274,168,150),(74,283,169,159),(75,292,170,88),(76,301,171,97),(77,310,172,106),(78,319,173,115),(79,248,174,124),(80,257,175,133)]])
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 | ··· | 16H | 16I | ··· | 16P | 20A | ··· | 20H | 40A | ··· | 40P | 80A | ··· | 80AF |
| 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 | 16 | ··· | 16 | 20 | ··· | 20 | 40 | ··· | 40 | 80 | ··· | 80 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 10 | 10 | 10 | 10 | 2 | 2 | 1 | ··· | 1 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 2 | ··· | 2 | 10 | ··· | 10 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
104 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | C8 | C8 | D5 | Dic5 | D10 | M5(2) | C4×D5 | C4×D5 | C8×D5 | C8×D5 | C80⋊C2 |
| kernel | C80⋊17C4 | C2×C5⋊2C16 | C8×Dic5 | C2×C80 | C5⋊2C16 | C80 | C2×C5⋊2C8 | C4×Dic5 | C5⋊2C8 | C2×Dic5 | C2×C16 | C16 | C2×C8 | C10 | C8 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 8 | 8 | 2 | 4 | 2 | 8 | 4 | 4 | 8 | 8 | 32 |
Matrix representation of C80⋊17C4 ►in GL3(𝔽241) generated by
| 177 | 0 | 0 |
| 0 | 125 | 117 |
| 0 | 124 | 66 |
| 64 | 0 | 0 |
| 0 | 20 | 29 |
| 0 | 194 | 221 |
G:=sub<GL(3,GF(241))| [177,0,0,0,125,124,0,117,66],[64,0,0,0,20,194,0,29,221] >;
C80⋊17C4 in GAP, Magma, Sage, TeX
C_{80}\rtimes_{17}C_4 % in TeX
G:=Group("C80:17C4"); // GroupNames label
G:=SmallGroup(320,60);
// by ID
G=gap.SmallGroup(320,60);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,477,64,80,102,12550]);
// Polycyclic
G:=Group<a,b|a^80=b^4=1,b*a*b^-1=a^9>;
// generators/relations
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