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G = C8×Dic10order 320 = 26·5

Direct product of C8 and Dic10

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C8×Dic10
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C4×Dic5 — C4×Dic10 — C8×Dic10
 Lower central C5 — C10 — C8×Dic10
 Upper central C1 — C2×C8 — C4×C8

Generators and relations for C8×Dic10
G = < a,b,c | a8=b20=1, c2=b10, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 254 in 102 conjugacy classes, 63 normal (33 characteristic)
C1, C2, C4, C4, C4, C22, C5, C8, C8, C2×C4, C2×C4, Q8, C10, C42, C42, C4⋊C4, C2×C8, C2×C8, C2×Q8, Dic5, Dic5, C20, C20, C20, C2×C10, C4×C8, C4×C8, C4⋊C8, C4×Q8, C52C8, C40, C40, Dic10, C2×Dic5, C2×C20, C8×Q8, C2×C52C8, C4×Dic5, C10.D4, C4⋊Dic5, C4×C20, C2×C40, C2×Dic10, C203C8, C8×Dic5, C20.8Q8, C4×C40, C4×Dic10, C8×Dic10
Quotients: C1, C2, C4, C22, C8, C2×C4, Q8, C23, D5, C2×C8, C22×C4, C2×Q8, C4○D4, D10, C4×Q8, C22×C8, C8○D4, Dic10, C4×D5, C22×D5, C8×Q8, C8×D5, C2×Dic10, C2×C4×D5, C4○D20, C4×Dic10, D5×C2×C8, D20.3C4, C8×Dic10

Smallest permutation representation of C8×Dic10
Regular action on 320 points
Generators in S320
(1 123 311 242 300 93 187 118)(2 124 312 243 281 94 188 119)(3 125 313 244 282 95 189 120)(4 126 314 245 283 96 190 101)(5 127 315 246 284 97 191 102)(6 128 316 247 285 98 192 103)(7 129 317 248 286 99 193 104)(8 130 318 249 287 100 194 105)(9 131 319 250 288 81 195 106)(10 132 320 251 289 82 196 107)(11 133 301 252 290 83 197 108)(12 134 302 253 291 84 198 109)(13 135 303 254 292 85 199 110)(14 136 304 255 293 86 200 111)(15 137 305 256 294 87 181 112)(16 138 306 257 295 88 182 113)(17 139 307 258 296 89 183 114)(18 140 308 259 297 90 184 115)(19 121 309 260 298 91 185 116)(20 122 310 241 299 92 186 117)(21 212 144 228 173 47 277 77)(22 213 145 229 174 48 278 78)(23 214 146 230 175 49 279 79)(24 215 147 231 176 50 280 80)(25 216 148 232 177 51 261 61)(26 217 149 233 178 52 262 62)(27 218 150 234 179 53 263 63)(28 219 151 235 180 54 264 64)(29 220 152 236 161 55 265 65)(30 201 153 237 162 56 266 66)(31 202 154 238 163 57 267 67)(32 203 155 239 164 58 268 68)(33 204 156 240 165 59 269 69)(34 205 157 221 166 60 270 70)(35 206 158 222 167 41 271 71)(36 207 159 223 168 42 272 72)(37 208 160 224 169 43 273 73)(38 209 141 225 170 44 274 74)(39 210 142 226 171 45 275 75)(40 211 143 227 172 46 276 76)
(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)
(1 224 11 234)(2 223 12 233)(3 222 13 232)(4 221 14 231)(5 240 15 230)(6 239 16 229)(7 238 17 228)(8 237 18 227)(9 236 19 226)(10 235 20 225)(21 99 31 89)(22 98 32 88)(23 97 33 87)(24 96 34 86)(25 95 35 85)(26 94 36 84)(27 93 37 83)(28 92 38 82)(29 91 39 81)(30 90 40 100)(41 303 51 313)(42 302 52 312)(43 301 53 311)(44 320 54 310)(45 319 55 309)(46 318 56 308)(47 317 57 307)(48 316 58 306)(49 315 59 305)(50 314 60 304)(61 282 71 292)(62 281 72 291)(63 300 73 290)(64 299 74 289)(65 298 75 288)(66 297 76 287)(67 296 77 286)(68 295 78 285)(69 294 79 284)(70 293 80 283)(101 157 111 147)(102 156 112 146)(103 155 113 145)(104 154 114 144)(105 153 115 143)(106 152 116 142)(107 151 117 141)(108 150 118 160)(109 149 119 159)(110 148 120 158)(121 171 131 161)(122 170 132 180)(123 169 133 179)(124 168 134 178)(125 167 135 177)(126 166 136 176)(127 165 137 175)(128 164 138 174)(129 163 139 173)(130 162 140 172)(181 214 191 204)(182 213 192 203)(183 212 193 202)(184 211 194 201)(185 210 195 220)(186 209 196 219)(187 208 197 218)(188 207 198 217)(189 206 199 216)(190 205 200 215)(241 274 251 264)(242 273 252 263)(243 272 253 262)(244 271 254 261)(245 270 255 280)(246 269 256 279)(247 268 257 278)(248 267 258 277)(249 266 259 276)(250 265 260 275)

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

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

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

104 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 4G 4H 4I ··· 4P 5A 5B 8A ··· 8H 8I 8J 8K 8L 8M ··· 8T 10A ··· 10F 20A ··· 20X 40A ··· 40AF order 1 2 2 2 4 4 4 4 4 4 4 4 4 ··· 4 5 5 8 ··· 8 8 8 8 8 8 ··· 8 10 ··· 10 20 ··· 20 40 ··· 40 size 1 1 1 1 1 1 1 1 2 2 2 2 10 ··· 10 2 2 1 ··· 1 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 2 2 type + + + + + + - + + + - image C1 C2 C2 C2 C2 C2 C4 C4 C4 C8 Q8 D5 C4○D4 D10 D10 C8○D4 Dic10 C4×D5 C8×D5 C4○D20 D20.3C4 kernel C8×Dic10 C20⋊3C8 C8×Dic5 C20.8Q8 C4×C40 C4×Dic10 C10.D4 C4⋊Dic5 C2×Dic10 Dic10 C40 C4×C8 C20 C42 C2×C8 C10 C8 C2×C4 C4 C4 C2 # reps 1 1 2 2 1 1 4 2 2 16 2 2 2 2 4 4 8 8 16 8 16

Matrix representation of C8×Dic10 in GL4(𝔽41) generated by

 27 0 0 0 0 27 0 0 0 0 3 0 0 0 0 3
,
 0 1 0 0 40 34 0 0 0 0 9 11 0 0 30 14
,
 27 14 0 0 30 14 0 0 0 0 0 32 0 0 32 0
G:=sub<GL(4,GF(41))| [27,0,0,0,0,27,0,0,0,0,3,0,0,0,0,3],[0,40,0,0,1,34,0,0,0,0,9,30,0,0,11,14],[27,30,0,0,14,14,0,0,0,0,0,32,0,0,32,0] >;

C8×Dic10 in GAP, Magma, Sage, TeX

C_8\times {\rm Dic}_{10}
% in TeX

G:=Group("C8xDic10");
// GroupNames label

G:=SmallGroup(320,305);
// by ID

G=gap.SmallGroup(320,305);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,253,120,58,136,12550]);
// Polycyclic

G:=Group<a,b,c|a^8=b^20=1,c^2=b^10,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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