Copied to
clipboard

## G = C4×C80order 320 = 26·5

### Abelian group of type [4,80]

Aliases: C4×C80, SmallGroup(320,150)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4×C80
 Chief series C1 — C2 — C4 — C2×C4 — C2×C8 — C2×C40 — C2×C80 — C4×C80
 Lower central C1 — C4×C80
 Upper central 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
(1 310 161 82)(2 311 162 83)(3 312 163 84)(4 313 164 85)(5 314 165 86)(6 315 166 87)(7 316 167 88)(8 317 168 89)(9 318 169 90)(10 319 170 91)(11 320 171 92)(12 241 172 93)(13 242 173 94)(14 243 174 95)(15 244 175 96)(16 245 176 97)(17 246 177 98)(18 247 178 99)(19 248 179 100)(20 249 180 101)(21 250 181 102)(22 251 182 103)(23 252 183 104)(24 253 184 105)(25 254 185 106)(26 255 186 107)(27 256 187 108)(28 257 188 109)(29 258 189 110)(30 259 190 111)(31 260 191 112)(32 261 192 113)(33 262 193 114)(34 263 194 115)(35 264 195 116)(36 265 196 117)(37 266 197 118)(38 267 198 119)(39 268 199 120)(40 269 200 121)(41 270 201 122)(42 271 202 123)(43 272 203 124)(44 273 204 125)(45 274 205 126)(46 275 206 127)(47 276 207 128)(48 277 208 129)(49 278 209 130)(50 279 210 131)(51 280 211 132)(52 281 212 133)(53 282 213 134)(54 283 214 135)(55 284 215 136)(56 285 216 137)(57 286 217 138)(58 287 218 139)(59 288 219 140)(60 289 220 141)(61 290 221 142)(62 291 222 143)(63 292 223 144)(64 293 224 145)(65 294 225 146)(66 295 226 147)(67 296 227 148)(68 297 228 149)(69 298 229 150)(70 299 230 151)(71 300 231 152)(72 301 232 153)(73 302 233 154)(74 303 234 155)(75 304 235 156)(76 305 236 157)(77 306 237 158)(78 307 238 159)(79 308 239 160)(80 309 240 81)
(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)| (1,310,161,82)(2,311,162,83)(3,312,163,84)(4,313,164,85)(5,314,165,86)(6,315,166,87)(7,316,167,88)(8,317,168,89)(9,318,169,90)(10,319,170,91)(11,320,171,92)(12,241,172,93)(13,242,173,94)(14,243,174,95)(15,244,175,96)(16,245,176,97)(17,246,177,98)(18,247,178,99)(19,248,179,100)(20,249,180,101)(21,250,181,102)(22,251,182,103)(23,252,183,104)(24,253,184,105)(25,254,185,106)(26,255,186,107)(27,256,187,108)(28,257,188,109)(29,258,189,110)(30,259,190,111)(31,260,191,112)(32,261,192,113)(33,262,193,114)(34,263,194,115)(35,264,195,116)(36,265,196,117)(37,266,197,118)(38,267,198,119)(39,268,199,120)(40,269,200,121)(41,270,201,122)(42,271,202,123)(43,272,203,124)(44,273,204,125)(45,274,205,126)(46,275,206,127)(47,276,207,128)(48,277,208,129)(49,278,209,130)(50,279,210,131)(51,280,211,132)(52,281,212,133)(53,282,213,134)(54,283,214,135)(55,284,215,136)(56,285,216,137)(57,286,217,138)(58,287,218,139)(59,288,219,140)(60,289,220,141)(61,290,221,142)(62,291,222,143)(63,292,223,144)(64,293,224,145)(65,294,225,146)(66,295,226,147)(67,296,227,148)(68,297,228,149)(69,298,229,150)(70,299,230,151)(71,300,231,152)(72,301,232,153)(73,302,233,154)(74,303,234,155)(75,304,235,156)(76,305,236,157)(77,306,237,158)(78,307,238,159)(79,308,239,160)(80,309,240,81), (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( (1,310,161,82)(2,311,162,83)(3,312,163,84)(4,313,164,85)(5,314,165,86)(6,315,166,87)(7,316,167,88)(8,317,168,89)(9,318,169,90)(10,319,170,91)(11,320,171,92)(12,241,172,93)(13,242,173,94)(14,243,174,95)(15,244,175,96)(16,245,176,97)(17,246,177,98)(18,247,178,99)(19,248,179,100)(20,249,180,101)(21,250,181,102)(22,251,182,103)(23,252,183,104)(24,253,184,105)(25,254,185,106)(26,255,186,107)(27,256,187,108)(28,257,188,109)(29,258,189,110)(30,259,190,111)(31,260,191,112)(32,261,192,113)(33,262,193,114)(34,263,194,115)(35,264,195,116)(36,265,196,117)(37,266,197,118)(38,267,198,119)(39,268,199,120)(40,269,200,121)(41,270,201,122)(42,271,202,123)(43,272,203,124)(44,273,204,125)(45,274,205,126)(46,275,206,127)(47,276,207,128)(48,277,208,129)(49,278,209,130)(50,279,210,131)(51,280,211,132)(52,281,212,133)(53,282,213,134)(54,283,214,135)(55,284,215,136)(56,285,216,137)(57,286,217,138)(58,287,218,139)(59,288,219,140)(60,289,220,141)(61,290,221,142)(62,291,222,143)(63,292,223,144)(64,293,224,145)(65,294,225,146)(66,295,226,147)(67,296,227,148)(68,297,228,149)(69,298,229,150)(70,299,230,151)(71,300,231,152)(72,301,232,153)(73,302,233,154)(74,303,234,155)(75,304,235,156)(76,305,236,157)(77,306,237,158)(78,307,238,159)(79,308,239,160)(80,309,240,81), (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([(1,310,161,82),(2,311,162,83),(3,312,163,84),(4,313,164,85),(5,314,165,86),(6,315,166,87),(7,316,167,88),(8,317,168,89),(9,318,169,90),(10,319,170,91),(11,320,171,92),(12,241,172,93),(13,242,173,94),(14,243,174,95),(15,244,175,96),(16,245,176,97),(17,246,177,98),(18,247,178,99),(19,248,179,100),(20,249,180,101),(21,250,181,102),(22,251,182,103),(23,252,183,104),(24,253,184,105),(25,254,185,106),(26,255,186,107),(27,256,187,108),(28,257,188,109),(29,258,189,110),(30,259,190,111),(31,260,191,112),(32,261,192,113),(33,262,193,114),(34,263,194,115),(35,264,195,116),(36,265,196,117),(37,266,197,118),(38,267,198,119),(39,268,199,120),(40,269,200,121),(41,270,201,122),(42,271,202,123),(43,272,203,124),(44,273,204,125),(45,274,205,126),(46,275,206,127),(47,276,207,128),(48,277,208,129),(49,278,209,130),(50,279,210,131),(51,280,211,132),(52,281,212,133),(53,282,213,134),(54,283,214,135),(55,284,215,136),(56,285,216,137),(57,286,217,138),(58,287,218,139),(59,288,219,140),(60,289,220,141),(61,290,221,142),(62,291,222,143),(63,292,223,144),(64,293,224,145),(65,294,225,146),(66,295,226,147),(67,296,227,148),(68,297,228,149),(69,298,229,150),(70,299,230,151),(71,300,231,152),(72,301,232,153),(73,302,233,154),(74,303,234,155),(75,304,235,156),(76,305,236,157),(77,306,237,158),(78,307,238,159),(79,308,239,160),(80,309,240,81)], [(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

Export

׿
×
𝔽