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## G = Dic10⋊3C8order 320 = 26·5

### 1st semidirect product of Dic10 and C8 acting via C8/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — Dic10⋊3C8
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — C4×C20 — C20⋊3C8 — Dic10⋊3C8
 Lower central C5 — C10 — C20 — Dic10⋊3C8
 Upper central C1 — C2×C4 — C42 — C4×C8

Generators and relations for Dic103C8
G = < a,b,c | a20=c8=1, b2=a10, bab-1=a-1, ac=ca, cbc-1=a15b >

Subgroups: 230 in 70 conjugacy classes, 35 normal (33 characteristic)
C1, C2 [×3], C4 [×4], C4 [×4], C22, C5, C8 [×3], C2×C4 [×3], C2×C4 [×2], Q8 [×3], C10 [×3], C42, C42, C4⋊C4 [×2], C2×C8 [×2], C2×Q8, Dic5 [×3], C20 [×4], C20, C2×C10, C4×C8, C4⋊C8, C4×Q8, C52C8, C40 [×2], Dic10 [×2], Dic10, C2×Dic5 [×2], C2×C20 [×3], Q8⋊C8, C2×C52C8, C4×Dic5, C10.D4, C4⋊Dic5, C4×C20, C2×C40, C2×Dic10, C203C8, C4×C40, C4×Dic10, Dic103C8
Quotients: C1, C2 [×3], C4 [×2], C22, C8 [×2], C2×C4, D4 [×2], D5, C22⋊C4, C2×C8, M4(2), SD16, Q16, D10, C22⋊C8, Q8⋊C4, C4≀C2, C4×D5, D20, C5⋊D4, Q8⋊C8, C8×D5, C8⋊D5, C40⋊C2, Dic20, D10⋊C4, D204C4, C20.44D4, D101C8, Dic103C8

Smallest permutation representation of Dic103C8
Regular action on 320 points
Generators in S320
(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)
(1 244 11 254)(2 243 12 253)(3 242 13 252)(4 241 14 251)(5 260 15 250)(6 259 16 249)(7 258 17 248)(8 257 18 247)(9 256 19 246)(10 255 20 245)(21 309 31 319)(22 308 32 318)(23 307 33 317)(24 306 34 316)(25 305 35 315)(26 304 36 314)(27 303 37 313)(28 302 38 312)(29 301 39 311)(30 320 40 310)(41 174 51 164)(42 173 52 163)(43 172 53 162)(44 171 54 161)(45 170 55 180)(46 169 56 179)(47 168 57 178)(48 167 58 177)(49 166 59 176)(50 165 60 175)(61 118 71 108)(62 117 72 107)(63 116 73 106)(64 115 74 105)(65 114 75 104)(66 113 76 103)(67 112 77 102)(68 111 78 101)(69 110 79 120)(70 109 80 119)(81 276 91 266)(82 275 92 265)(83 274 93 264)(84 273 94 263)(85 272 95 262)(86 271 96 261)(87 270 97 280)(88 269 98 279)(89 268 99 278)(90 267 100 277)(121 181 131 191)(122 200 132 190)(123 199 133 189)(124 198 134 188)(125 197 135 187)(126 196 136 186)(127 195 137 185)(128 194 138 184)(129 193 139 183)(130 192 140 182)(141 288 151 298)(142 287 152 297)(143 286 153 296)(144 285 154 295)(145 284 155 294)(146 283 156 293)(147 282 157 292)(148 281 158 291)(149 300 159 290)(150 299 160 289)(201 240 211 230)(202 239 212 229)(203 238 213 228)(204 237 214 227)(205 236 215 226)(206 235 216 225)(207 234 217 224)(208 233 218 223)(209 232 219 222)(210 231 220 221)
(1 224 282 274 185 105 37 164)(2 225 283 275 186 106 38 165)(3 226 284 276 187 107 39 166)(4 227 285 277 188 108 40 167)(5 228 286 278 189 109 21 168)(6 229 287 279 190 110 22 169)(7 230 288 280 191 111 23 170)(8 231 289 261 192 112 24 171)(9 232 290 262 193 113 25 172)(10 233 291 263 194 114 26 173)(11 234 292 264 195 115 27 174)(12 235 293 265 196 116 28 175)(13 236 294 266 197 117 29 176)(14 237 295 267 198 118 30 177)(15 238 296 268 199 119 31 178)(16 239 297 269 200 120 32 179)(17 240 298 270 181 101 33 180)(18 221 299 271 182 102 34 161)(19 222 300 272 183 103 35 162)(20 223 281 273 184 104 36 163)(41 259 217 142 93 122 74 318)(42 260 218 143 94 123 75 319)(43 241 219 144 95 124 76 320)(44 242 220 145 96 125 77 301)(45 243 201 146 97 126 78 302)(46 244 202 147 98 127 79 303)(47 245 203 148 99 128 80 304)(48 246 204 149 100 129 61 305)(49 247 205 150 81 130 62 306)(50 248 206 151 82 131 63 307)(51 249 207 152 83 132 64 308)(52 250 208 153 84 133 65 309)(53 251 209 154 85 134 66 310)(54 252 210 155 86 135 67 311)(55 253 211 156 87 136 68 312)(56 254 212 157 88 137 69 313)(57 255 213 158 89 138 70 314)(58 256 214 159 90 139 71 315)(59 257 215 160 91 140 72 316)(60 258 216 141 92 121 73 317)

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

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

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

92 conjugacy classes

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

92 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + - + + - image C1 C2 C2 C2 C4 C4 C8 D4 D5 M4(2) SD16 Q16 D10 C4≀C2 C4×D5 D20 C5⋊D4 C8×D5 C8⋊D5 C40⋊C2 Dic20 D20⋊4C4 kernel Dic10⋊3C8 C20⋊3C8 C4×C40 C4×Dic10 C4⋊Dic5 C2×Dic10 Dic10 C2×C20 C4×C8 C20 C20 C20 C42 C10 C2×C4 C2×C4 C2×C4 C4 C4 C4 C4 C2 # reps 1 1 1 1 2 2 8 2 2 2 2 2 2 4 4 4 4 8 8 8 8 16

Matrix representation of Dic103C8 in GL3(𝔽41) generated by

 1 0 0 0 21 0 0 0 2
,
 1 0 0 0 0 14 0 38 0
,
 3 0 0 0 27 0 0 0 38
G:=sub<GL(3,GF(41))| [1,0,0,0,21,0,0,0,2],[1,0,0,0,0,38,0,14,0],[3,0,0,0,27,0,0,0,38] >;

Dic103C8 in GAP, Magma, Sage, TeX

{\rm Dic}_{10}\rtimes_3C_8
% in TeX

G:=Group("Dic10:3C8");
// GroupNames label

G:=SmallGroup(320,14);
// by ID

G=gap.SmallGroup(320,14);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,85,92,422,100,136,12550]);
// Polycyclic

G:=Group<a,b,c|a^20=c^8=1,b^2=a^10,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^15*b>;
// generators/relations

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