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G = C8×C40order 320 = 26·5

Abelian group of type [8,40]

direct product, abelian, monomial, 2-elementary

Aliases: C8×C40, SmallGroup(320,126)

Series: Derived Chief Lower central Upper central

C1 — C8×C40
C1C2C22C2×C4C42C4×C20C4×C40 — C8×C40
C1 — C8×C40
C1 — C8×C40

Generators and relations for C8×C40
 G = < a,b | a8=b40=1, ab=ba >

Subgroups: 74, all normal (8 characteristic)
C1, C2 [×3], C4 [×6], C22, C5, C8 [×12], C2×C4 [×3], C10 [×3], C42, C2×C8 [×6], C20 [×6], C2×C10, C4×C8 [×3], C40 [×12], C2×C20 [×3], C82, C4×C20, C2×C40 [×6], C4×C40 [×3], C8×C40
Quotients: C1, C2 [×3], C4 [×6], C22, C5, C8 [×12], C2×C4 [×3], C10 [×3], C42, C2×C8 [×6], C20 [×6], C2×C10, C4×C8 [×3], C40 [×12], C2×C20 [×3], C82, C4×C20, C2×C40 [×6], C4×C40 [×3], C8×C40

Smallest permutation representation of C8×C40
Regular action on 320 points
Generators in S320
(1 280 293 144 172 116 212 63)(2 241 294 145 173 117 213 64)(3 242 295 146 174 118 214 65)(4 243 296 147 175 119 215 66)(5 244 297 148 176 120 216 67)(6 245 298 149 177 81 217 68)(7 246 299 150 178 82 218 69)(8 247 300 151 179 83 219 70)(9 248 301 152 180 84 220 71)(10 249 302 153 181 85 221 72)(11 250 303 154 182 86 222 73)(12 251 304 155 183 87 223 74)(13 252 305 156 184 88 224 75)(14 253 306 157 185 89 225 76)(15 254 307 158 186 90 226 77)(16 255 308 159 187 91 227 78)(17 256 309 160 188 92 228 79)(18 257 310 121 189 93 229 80)(19 258 311 122 190 94 230 41)(20 259 312 123 191 95 231 42)(21 260 313 124 192 96 232 43)(22 261 314 125 193 97 233 44)(23 262 315 126 194 98 234 45)(24 263 316 127 195 99 235 46)(25 264 317 128 196 100 236 47)(26 265 318 129 197 101 237 48)(27 266 319 130 198 102 238 49)(28 267 320 131 199 103 239 50)(29 268 281 132 200 104 240 51)(30 269 282 133 161 105 201 52)(31 270 283 134 162 106 202 53)(32 271 284 135 163 107 203 54)(33 272 285 136 164 108 204 55)(34 273 286 137 165 109 205 56)(35 274 287 138 166 110 206 57)(36 275 288 139 167 111 207 58)(37 276 289 140 168 112 208 59)(38 277 290 141 169 113 209 60)(39 278 291 142 170 114 210 61)(40 279 292 143 171 115 211 62)
(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)| (1,280,293,144,172,116,212,63)(2,241,294,145,173,117,213,64)(3,242,295,146,174,118,214,65)(4,243,296,147,175,119,215,66)(5,244,297,148,176,120,216,67)(6,245,298,149,177,81,217,68)(7,246,299,150,178,82,218,69)(8,247,300,151,179,83,219,70)(9,248,301,152,180,84,220,71)(10,249,302,153,181,85,221,72)(11,250,303,154,182,86,222,73)(12,251,304,155,183,87,223,74)(13,252,305,156,184,88,224,75)(14,253,306,157,185,89,225,76)(15,254,307,158,186,90,226,77)(16,255,308,159,187,91,227,78)(17,256,309,160,188,92,228,79)(18,257,310,121,189,93,229,80)(19,258,311,122,190,94,230,41)(20,259,312,123,191,95,231,42)(21,260,313,124,192,96,232,43)(22,261,314,125,193,97,233,44)(23,262,315,126,194,98,234,45)(24,263,316,127,195,99,235,46)(25,264,317,128,196,100,236,47)(26,265,318,129,197,101,237,48)(27,266,319,130,198,102,238,49)(28,267,320,131,199,103,239,50)(29,268,281,132,200,104,240,51)(30,269,282,133,161,105,201,52)(31,270,283,134,162,106,202,53)(32,271,284,135,163,107,203,54)(33,272,285,136,164,108,204,55)(34,273,286,137,165,109,205,56)(35,274,287,138,166,110,206,57)(36,275,288,139,167,111,207,58)(37,276,289,140,168,112,208,59)(38,277,290,141,169,113,209,60)(39,278,291,142,170,114,210,61)(40,279,292,143,171,115,211,62), (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( (1,280,293,144,172,116,212,63)(2,241,294,145,173,117,213,64)(3,242,295,146,174,118,214,65)(4,243,296,147,175,119,215,66)(5,244,297,148,176,120,216,67)(6,245,298,149,177,81,217,68)(7,246,299,150,178,82,218,69)(8,247,300,151,179,83,219,70)(9,248,301,152,180,84,220,71)(10,249,302,153,181,85,221,72)(11,250,303,154,182,86,222,73)(12,251,304,155,183,87,223,74)(13,252,305,156,184,88,224,75)(14,253,306,157,185,89,225,76)(15,254,307,158,186,90,226,77)(16,255,308,159,187,91,227,78)(17,256,309,160,188,92,228,79)(18,257,310,121,189,93,229,80)(19,258,311,122,190,94,230,41)(20,259,312,123,191,95,231,42)(21,260,313,124,192,96,232,43)(22,261,314,125,193,97,233,44)(23,262,315,126,194,98,234,45)(24,263,316,127,195,99,235,46)(25,264,317,128,196,100,236,47)(26,265,318,129,197,101,237,48)(27,266,319,130,198,102,238,49)(28,267,320,131,199,103,239,50)(29,268,281,132,200,104,240,51)(30,269,282,133,161,105,201,52)(31,270,283,134,162,106,202,53)(32,271,284,135,163,107,203,54)(33,272,285,136,164,108,204,55)(34,273,286,137,165,109,205,56)(35,274,287,138,166,110,206,57)(36,275,288,139,167,111,207,58)(37,276,289,140,168,112,208,59)(38,277,290,141,169,113,209,60)(39,278,291,142,170,114,210,61)(40,279,292,143,171,115,211,62), (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([(1,280,293,144,172,116,212,63),(2,241,294,145,173,117,213,64),(3,242,295,146,174,118,214,65),(4,243,296,147,175,119,215,66),(5,244,297,148,176,120,216,67),(6,245,298,149,177,81,217,68),(7,246,299,150,178,82,218,69),(8,247,300,151,179,83,219,70),(9,248,301,152,180,84,220,71),(10,249,302,153,181,85,221,72),(11,250,303,154,182,86,222,73),(12,251,304,155,183,87,223,74),(13,252,305,156,184,88,224,75),(14,253,306,157,185,89,225,76),(15,254,307,158,186,90,226,77),(16,255,308,159,187,91,227,78),(17,256,309,160,188,92,228,79),(18,257,310,121,189,93,229,80),(19,258,311,122,190,94,230,41),(20,259,312,123,191,95,231,42),(21,260,313,124,192,96,232,43),(22,261,314,125,193,97,233,44),(23,262,315,126,194,98,234,45),(24,263,316,127,195,99,235,46),(25,264,317,128,196,100,236,47),(26,265,318,129,197,101,237,48),(27,266,319,130,198,102,238,49),(28,267,320,131,199,103,239,50),(29,268,281,132,200,104,240,51),(30,269,282,133,161,105,201,52),(31,270,283,134,162,106,202,53),(32,271,284,135,163,107,203,54),(33,272,285,136,164,108,204,55),(34,273,286,137,165,109,205,56),(35,274,287,138,166,110,206,57),(36,275,288,139,167,111,207,58),(37,276,289,140,168,112,208,59),(38,277,290,141,169,113,209,60),(39,278,291,142,170,114,210,61),(40,279,292,143,171,115,211,62)], [(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 2A2B2C4A···4L5A5B5C5D8A···8AV10A···10L20A···20AV40A···40GJ
order12224···455558···810···1020···2040···40
size11111···111111···11···11···11···1

320 irreducible representations

dim11111111
type++
imageC1C2C4C5C8C10C20C40
kernelC8×C40C4×C40C2×C40C82C40C4×C8C2×C8C8
# reps13124481248192

Matrix representation of C8×C40 in GL2(𝔽41) generated by

380
01
,
120
013
G:=sub<GL(2,GF(41))| [38,0,0,1],[12,0,0,13] >;

C8×C40 in GAP, Magma, Sage, TeX

C_8\times C_{40}
% in TeX

G:=Group("C8xC40");
// GroupNames label

G:=SmallGroup(320,126);
// by ID

G=gap.SmallGroup(320,126);
# by ID

G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-2,140,288,136,172]);
// Polycyclic

G:=Group<a,b|a^8=b^40=1,a*b=b*a>;
// generators/relations

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