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## G = C8.6Dic10order 320 = 26·5

### 3rd non-split extension by C8 of Dic10 acting via Dic10/Dic5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — C8.6Dic10
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — C4×Dic5 — C8×Dic5 — C8.6Dic10
 Lower central C5 — C10 — C2×C20 — C8.6Dic10
 Upper central C1 — C22 — C2×C4 — C2.D8

Generators and relations for C8.6Dic10
G = < a,b,c | a8=b20=1, c2=b10, bab-1=a-1, ac=ca, cbc-1=a4b-1 >

Subgroups: 286 in 86 conjugacy classes, 43 normal (27 characteristic)
C1, C2, C4, C4, C22, C5, C8, C8, C2×C4, C2×C4, C10, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, Dic5, C20, C20, C2×C10, C4×C8, C4.Q8, C2.D8, C2.D8, C42.C2, C52C8, C40, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C8.5Q8, C2×C52C8, C4×Dic5, C10.D4, C4⋊Dic5, C4⋊Dic5, C5×C4⋊C4, C2×C40, C20.Q8, C8×Dic5, C405C4, C5×C2.D8, C4.Dic10, C8.6Dic10
Quotients: C1, C2, C22, D4, Q8, C23, D5, C2×D4, C2×Q8, D10, C4⋊Q8, C4○D8, Dic10, C22×D5, C8.5Q8, C2×Dic10, D4×D5, Q8×D5, C20⋊Q8, D83D5, Q8.D10, C8.6Dic10

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

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

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

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

50 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 5A 5B 8A 8B 8C 8D 8E 8F 8G 8H 10A ··· 10F 20A 20B 20C 20D 20E ··· 20L 40A ··· 40H 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 20 20 20 ··· 20 40 ··· 40 size 1 1 1 1 2 2 8 8 10 10 10 10 40 40 2 2 2 2 2 2 10 10 10 10 2 ··· 2 4 4 4 4 8 ··· 8 4 ··· 4

50 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + - - + + + + - - + - + image C1 C2 C2 C2 C2 C2 Q8 Q8 D4 D5 D10 D10 C4○D8 Dic10 Q8×D5 D4×D5 D8⋊3D5 Q8.D10 kernel C8.6Dic10 C20.Q8 C8×Dic5 C40⋊5C4 C5×C2.D8 C4.Dic10 C5⋊2C8 C40 C2×Dic5 C2.D8 C4⋊C4 C2×C8 C10 C8 C4 C22 C2 C2 # reps 1 2 1 1 1 2 2 2 2 2 4 2 8 8 2 2 4 4

Matrix representation of C8.6Dic10 in GL4(𝔽41) generated by

 17 24 0 0 29 0 0 0 0 0 1 0 0 0 0 1
,
 37 12 0 0 2 4 0 0 0 0 25 2 0 0 39 13
,
 9 0 0 0 0 9 0 0 0 0 20 20 0 0 23 21
G:=sub<GL(4,GF(41))| [17,29,0,0,24,0,0,0,0,0,1,0,0,0,0,1],[37,2,0,0,12,4,0,0,0,0,25,39,0,0,2,13],[9,0,0,0,0,9,0,0,0,0,20,23,0,0,20,21] >;

C8.6Dic10 in GAP, Magma, Sage, TeX

C_8._6{\rm Dic}_{10}
% in TeX

G:=Group("C8.6Dic10");
// GroupNames label

G:=SmallGroup(320,505);
// by ID

G=gap.SmallGroup(320,505);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,120,926,219,58,438,102,12550]);
// Polycyclic

G:=Group<a,b,c|a^8=b^20=1,c^2=b^10,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^4*b^-1>;
// generators/relations

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