direct product, abelian, monomial, 2-elementary
Aliases: C4×C80, SmallGroup(320,150)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C4×C80 |
| C1 — C4×C80 |
| C1 — C4×C80 |
Generators and relations for C4×C80
G = < a,b | a4=b80=1, ab=ba >
Smallest permutation representation of C4×C80
►Regular action on 320 points
Generators in S320
(1 143 223 279)(2 144 224 280)(3 145 225 281)(4 146 226 282)(5 147 227 283)(6 148 228 284)(7 149 229 285)(8 150 230 286)(9 151 231 287)(10 152 232 288)(11 153 233 289)(12 154 234 290)(13 155 235 291)(14 156 236 292)(15 157 237 293)(16 158 238 294)(17 159 239 295)(18 160 240 296)(19 81 161 297)(20 82 162 298)(21 83 163 299)(22 84 164 300)(23 85 165 301)(24 86 166 302)(25 87 167 303)(26 88 168 304)(27 89 169 305)(28 90 170 306)(29 91 171 307)(30 92 172 308)(31 93 173 309)(32 94 174 310)(33 95 175 311)(34 96 176 312)(35 97 177 313)(36 98 178 314)(37 99 179 315)(38 100 180 316)(39 101 181 317)(40 102 182 318)(41 103 183 319)(42 104 184 320)(43 105 185 241)(44 106 186 242)(45 107 187 243)(46 108 188 244)(47 109 189 245)(48 110 190 246)(49 111 191 247)(50 112 192 248)(51 113 193 249)(52 114 194 250)(53 115 195 251)(54 116 196 252)(55 117 197 253)(56 118 198 254)(57 119 199 255)(58 120 200 256)(59 121 201 257)(60 122 202 258)(61 123 203 259)(62 124 204 260)(63 125 205 261)(64 126 206 262)(65 127 207 263)(66 128 208 264)(67 129 209 265)(68 130 210 266)(69 131 211 267)(70 132 212 268)(71 133 213 269)(72 134 214 270)(73 135 215 271)(74 136 216 272)(75 137 217 273)(76 138 218 274)(77 139 219 275)(78 140 220 276)(79 141 221 277)(80 142 222 278)
(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)
G:=sub<Sym(320)| (1,143,223,279)(2,144,224,280)(3,145,225,281)(4,146,226,282)(5,147,227,283)(6,148,228,284)(7,149,229,285)(8,150,230,286)(9,151,231,287)(10,152,232,288)(11,153,233,289)(12,154,234,290)(13,155,235,291)(14,156,236,292)(15,157,237,293)(16,158,238,294)(17,159,239,295)(18,160,240,296)(19,81,161,297)(20,82,162,298)(21,83,163,299)(22,84,164,300)(23,85,165,301)(24,86,166,302)(25,87,167,303)(26,88,168,304)(27,89,169,305)(28,90,170,306)(29,91,171,307)(30,92,172,308)(31,93,173,309)(32,94,174,310)(33,95,175,311)(34,96,176,312)(35,97,177,313)(36,98,178,314)(37,99,179,315)(38,100,180,316)(39,101,181,317)(40,102,182,318)(41,103,183,319)(42,104,184,320)(43,105,185,241)(44,106,186,242)(45,107,187,243)(46,108,188,244)(47,109,189,245)(48,110,190,246)(49,111,191,247)(50,112,192,248)(51,113,193,249)(52,114,194,250)(53,115,195,251)(54,116,196,252)(55,117,197,253)(56,118,198,254)(57,119,199,255)(58,120,200,256)(59,121,201,257)(60,122,202,258)(61,123,203,259)(62,124,204,260)(63,125,205,261)(64,126,206,262)(65,127,207,263)(66,128,208,264)(67,129,209,265)(68,130,210,266)(69,131,211,267)(70,132,212,268)(71,133,213,269)(72,134,214,270)(73,135,215,271)(74,136,216,272)(75,137,217,273)(76,138,218,274)(77,139,219,275)(78,140,220,276)(79,141,221,277)(80,142,222,278), (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)>;
G:=Group( (1,143,223,279)(2,144,224,280)(3,145,225,281)(4,146,226,282)(5,147,227,283)(6,148,228,284)(7,149,229,285)(8,150,230,286)(9,151,231,287)(10,152,232,288)(11,153,233,289)(12,154,234,290)(13,155,235,291)(14,156,236,292)(15,157,237,293)(16,158,238,294)(17,159,239,295)(18,160,240,296)(19,81,161,297)(20,82,162,298)(21,83,163,299)(22,84,164,300)(23,85,165,301)(24,86,166,302)(25,87,167,303)(26,88,168,304)(27,89,169,305)(28,90,170,306)(29,91,171,307)(30,92,172,308)(31,93,173,309)(32,94,174,310)(33,95,175,311)(34,96,176,312)(35,97,177,313)(36,98,178,314)(37,99,179,315)(38,100,180,316)(39,101,181,317)(40,102,182,318)(41,103,183,319)(42,104,184,320)(43,105,185,241)(44,106,186,242)(45,107,187,243)(46,108,188,244)(47,109,189,245)(48,110,190,246)(49,111,191,247)(50,112,192,248)(51,113,193,249)(52,114,194,250)(53,115,195,251)(54,116,196,252)(55,117,197,253)(56,118,198,254)(57,119,199,255)(58,120,200,256)(59,121,201,257)(60,122,202,258)(61,123,203,259)(62,124,204,260)(63,125,205,261)(64,126,206,262)(65,127,207,263)(66,128,208,264)(67,129,209,265)(68,130,210,266)(69,131,211,267)(70,132,212,268)(71,133,213,269)(72,134,214,270)(73,135,215,271)(74,136,216,272)(75,137,217,273)(76,138,218,274)(77,139,219,275)(78,140,220,276)(79,141,221,277)(80,142,222,278), (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) );
G=PermutationGroup([[(1,143,223,279),(2,144,224,280),(3,145,225,281),(4,146,226,282),(5,147,227,283),(6,148,228,284),(7,149,229,285),(8,150,230,286),(9,151,231,287),(10,152,232,288),(11,153,233,289),(12,154,234,290),(13,155,235,291),(14,156,236,292),(15,157,237,293),(16,158,238,294),(17,159,239,295),(18,160,240,296),(19,81,161,297),(20,82,162,298),(21,83,163,299),(22,84,164,300),(23,85,165,301),(24,86,166,302),(25,87,167,303),(26,88,168,304),(27,89,169,305),(28,90,170,306),(29,91,171,307),(30,92,172,308),(31,93,173,309),(32,94,174,310),(33,95,175,311),(34,96,176,312),(35,97,177,313),(36,98,178,314),(37,99,179,315),(38,100,180,316),(39,101,181,317),(40,102,182,318),(41,103,183,319),(42,104,184,320),(43,105,185,241),(44,106,186,242),(45,107,187,243),(46,108,188,244),(47,109,189,245),(48,110,190,246),(49,111,191,247),(50,112,192,248),(51,113,193,249),(52,114,194,250),(53,115,195,251),(54,116,196,252),(55,117,197,253),(56,118,198,254),(57,119,199,255),(58,120,200,256),(59,121,201,257),(60,122,202,258),(61,123,203,259),(62,124,204,260),(63,125,205,261),(64,126,206,262),(65,127,207,263),(66,128,208,264),(67,129,209,265),(68,130,210,266),(69,131,211,267),(70,132,212,268),(71,133,213,269),(72,134,214,270),(73,135,215,271),(74,136,216,272),(75,137,217,273),(76,138,218,274),(77,139,219,275),(78,140,220,276),(79,141,221,277),(80,142,222,278)], [(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)]])
320 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4L | 5A | 5B | 5C | 5D | 8A | ··· | 8P | 10A | ··· | 10L | 16A | ··· | 16AF | 20A | ··· | 20AV | 40A | ··· | 40BL | 80A | ··· | 80DX |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 16 | ··· | 16 | 20 | ··· | 20 | 40 | ··· | 40 | 80 | ··· | 80 |
| size | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
320 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | + | |||||||||||||||
| image | C1 | C2 | C2 | C4 | C4 | C4 | C5 | C8 | C8 | C10 | C10 | C16 | C20 | C20 | C20 | C40 | C40 | C80 |
| kernel | C4×C80 | C4×C40 | C2×C80 | C80 | C4×C20 | C2×C40 | C4×C16 | C40 | C2×C20 | C4×C8 | C2×C16 | C20 | C16 | C42 | C2×C8 | C8 | C2×C4 | C4 |
| # reps | 1 | 1 | 2 | 8 | 2 | 2 | 4 | 8 | 8 | 4 | 8 | 32 | 32 | 8 | 8 | 32 | 32 | 128 |
Matrix representation of C4×C80 ►in GL2(𝔽241) generated by
| 177 | 0 |
| 0 | 240 |
| 148 | 0 |
| 0 | 215 |
G:=sub<GL(2,GF(241))| [177,0,0,240],[148,0,0,215] >;
C4×C80 in GAP, Magma, Sage, TeX
C_4\times C_{80} % in TeX
G:=Group("C4xC80"); // GroupNames label
G:=SmallGroup(320,150);
// by ID
G=gap.SmallGroup(320,150);
# by ID
G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-2,140,288,136,124]);
// Polycyclic
G:=Group<a,b|a^4=b^80=1,a*b=b*a>;
// generators/relations
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