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## G = C8.8Dic10order 320 = 26·5

### 5th non-split extension by C8 of Dic10 acting via Dic10/Dic5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — C8.8Dic10
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — C4×Dic5 — C8×Dic5 — C8.8Dic10
 Lower central C5 — C10 — C2×C20 — C8.8Dic10
 Upper central C1 — C22 — C2×C4 — C4.Q8

Generators and relations for C8.8Dic10
G = < a,b,c | a8=b20=1, c2=b10, bab-1=a3, ac=ca, cbc-1=a4b-1 >

Subgroups: 286 in 86 conjugacy classes, 43 normal (21 characteristic)
C1, C2, C2, C4, C4, C22, C5, C8, C8, C2×C4, C2×C4, C10, C10, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, Dic5, C20, C20, C2×C10, C4×C8, C4.Q8, C4.Q8, 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, C10.D8, C8×Dic5, C406C4, C5×C4.Q8, C4.Dic10, C8.8Dic10
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, SD163D5, C8.8Dic10

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

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

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

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

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 type + + + + + + - - + + + + - - + image C1 C2 C2 C2 C2 C2 Q8 Q8 D4 D5 D10 D10 C4○D8 Dic10 Q8×D5 D4×D5 SD16⋊3D5 kernel C8.8Dic10 C10.D8 C8×Dic5 C40⋊6C4 C5×C4.Q8 C4.Dic10 C5⋊2C8 C40 C2×Dic5 C4.Q8 C4⋊C4 C2×C8 C10 C8 C4 C22 C2 # reps 1 2 1 1 1 2 2 2 2 2 4 2 8 8 2 2 8

Matrix representation of C8.8Dic10 in GL4(𝔽41) generated by

 15 26 0 0 15 15 0 0 0 0 40 0 0 0 0 40
,
 14 16 0 0 16 27 0 0 0 0 16 39 0 0 2 28
,
 0 9 0 0 32 0 0 0 0 0 18 6 0 0 21 23
G:=sub<GL(4,GF(41))| [15,15,0,0,26,15,0,0,0,0,40,0,0,0,0,40],[14,16,0,0,16,27,0,0,0,0,16,2,0,0,39,28],[0,32,0,0,9,0,0,0,0,0,18,21,0,0,6,23] >;

C8.8Dic10 in GAP, Magma, Sage, TeX

C_8._8{\rm Dic}_{10}
% in TeX

G:=Group("C8.8Dic10");
// GroupNames label

G:=SmallGroup(320,485);
// by ID

G=gap.SmallGroup(320,485);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,120,926,555,58,438,102,12550]);
// Polycyclic

G:=Group<a,b,c|a^8=b^20=1,c^2=b^10,b*a*b^-1=a^3,a*c=c*a,c*b*c^-1=a^4*b^-1>;
// generators/relations

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