metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic10⋊2C8, C5⋊C8⋊4Q8, C5⋊1(C8×Q8), C2.1(Q8×F5), C20.7(C2×C8), C4⋊C4.13F5, C10.1(C4×Q8), C4.4(D5⋊C8), C20⋊C8.3C2, C10.9(C8○D4), C10.7(C22×C8), Dic5.1(C2×C8), C2.4(D4.F5), Dic5.28(C2×Q8), C10.D4.9C4, (C2×Dic10).11C4, Dic5⋊C8.4C2, Dic5.67(C4○D4), Dic5⋊3Q8.15C2, C22.38(C22×F5), (C2×Dic5).330C23, (C4×Dic5).191C22, (C4×C5⋊C8).8C2, (C5×C4⋊C4).7C4, C2.9(C2×D5⋊C8), (C2×C4).61(C2×F5), (C2×C20).93(C2×C4), (C2×C5⋊C8).27C22, (C2×C10).41(C22×C4), (C2×Dic5).56(C2×C4), SmallGroup(320,1041)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 282 in 102 conjugacy classes, 56 normal (26 characteristic)
C1, C2 [×3], C4 [×2], C4 [×9], C22, C5, C8 [×5], C2×C4, C2×C4 [×2], C2×C4 [×4], Q8 [×4], C10 [×3], C42 [×3], C4⋊C4, C4⋊C4 [×2], C2×C8 [×4], C2×Q8, Dic5 [×2], Dic5 [×4], Dic5, C20 [×2], C20 [×2], C2×C10, C4×C8 [×3], C4⋊C8 [×3], C4×Q8, C5⋊C8 [×2], C5⋊C8 [×3], Dic10 [×4], C2×Dic5 [×2], C2×Dic5 [×2], C2×C20, C2×C20 [×2], C8×Q8, C4×Dic5, C4×Dic5 [×2], C10.D4 [×2], C5×C4⋊C4, C2×C5⋊C8 [×2], C2×C5⋊C8 [×2], C2×Dic10, C4×C5⋊C8, C4×C5⋊C8 [×2], C20⋊C8, Dic5⋊C8 [×2], Dic5⋊3Q8, Dic10⋊C8
Quotients:
C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], Q8 [×2], C23, C2×C8 [×6], C22×C4, C2×Q8, C4○D4, F5, C4×Q8, C22×C8, C8○D4, C2×F5 [×3], C8×Q8, D5⋊C8 [×2], C22×F5, C2×D5⋊C8, D4.F5, Q8×F5, Dic10⋊C8
Generators and relations
G = < a,b,c | a20=c8=1, b2=a10, bab-1=a-1, cac-1=a13, bc=cb >
►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)
(1 183 11 193)(2 182 12 192)(3 181 13 191)(4 200 14 190)(5 199 15 189)(6 198 16 188)(7 197 17 187)(8 196 18 186)(9 195 19 185)(10 194 20 184)(21 103 31 113)(22 102 32 112)(23 101 33 111)(24 120 34 110)(25 119 35 109)(26 118 36 108)(27 117 37 107)(28 116 38 106)(29 115 39 105)(30 114 40 104)(41 161 51 171)(42 180 52 170)(43 179 53 169)(44 178 54 168)(45 177 55 167)(46 176 56 166)(47 175 57 165)(48 174 58 164)(49 173 59 163)(50 172 60 162)(61 261 71 271)(62 280 72 270)(63 279 73 269)(64 278 74 268)(65 277 75 267)(66 276 76 266)(67 275 77 265)(68 274 78 264)(69 273 79 263)(70 272 80 262)(81 247 91 257)(82 246 92 256)(83 245 93 255)(84 244 94 254)(85 243 95 253)(86 242 96 252)(87 241 97 251)(88 260 98 250)(89 259 99 249)(90 258 100 248)(121 155 131 145)(122 154 132 144)(123 153 133 143)(124 152 134 142)(125 151 135 141)(126 150 136 160)(127 149 137 159)(128 148 138 158)(129 147 139 157)(130 146 140 156)(201 301 211 311)(202 320 212 310)(203 319 213 309)(204 318 214 308)(205 317 215 307)(206 316 216 306)(207 315 217 305)(208 314 218 304)(209 313 219 303)(210 312 220 302)(221 282 231 292)(222 281 232 291)(223 300 233 290)(224 299 234 289)(225 298 235 288)(226 297 236 287)(227 296 237 286)(228 295 238 285)(229 294 239 284)(230 293 240 283)
(1 223 163 66 113 214 145 255)(2 240 172 79 114 211 154 248)(3 237 161 72 115 208 143 241)(4 234 170 65 116 205 152 254)(5 231 179 78 117 202 141 247)(6 228 168 71 118 219 150 260)(7 225 177 64 119 216 159 253)(8 222 166 77 120 213 148 246)(9 239 175 70 101 210 157 259)(10 236 164 63 102 207 146 252)(11 233 173 76 103 204 155 245)(12 230 162 69 104 201 144 258)(13 227 171 62 105 218 153 251)(14 224 180 75 106 215 142 244)(15 221 169 68 107 212 151 257)(16 238 178 61 108 209 160 250)(17 235 167 74 109 206 149 243)(18 232 176 67 110 203 158 256)(19 229 165 80 111 220 147 249)(20 226 174 73 112 217 156 242)(21 308 121 83 183 300 49 276)(22 305 130 96 184 297 58 269)(23 302 139 89 185 294 47 262)(24 319 128 82 186 291 56 275)(25 316 137 95 187 288 45 268)(26 313 126 88 188 285 54 261)(27 310 135 81 189 282 43 274)(28 307 124 94 190 299 52 267)(29 304 133 87 191 296 41 280)(30 301 122 100 192 293 50 273)(31 318 131 93 193 290 59 266)(32 315 140 86 194 287 48 279)(33 312 129 99 195 284 57 272)(34 309 138 92 196 281 46 265)(35 306 127 85 197 298 55 278)(36 303 136 98 198 295 44 271)(37 320 125 91 199 292 53 264)(38 317 134 84 200 289 42 277)(39 314 123 97 181 286 51 270)(40 311 132 90 182 283 60 263)
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), (1,183,11,193)(2,182,12,192)(3,181,13,191)(4,200,14,190)(5,199,15,189)(6,198,16,188)(7,197,17,187)(8,196,18,186)(9,195,19,185)(10,194,20,184)(21,103,31,113)(22,102,32,112)(23,101,33,111)(24,120,34,110)(25,119,35,109)(26,118,36,108)(27,117,37,107)(28,116,38,106)(29,115,39,105)(30,114,40,104)(41,161,51,171)(42,180,52,170)(43,179,53,169)(44,178,54,168)(45,177,55,167)(46,176,56,166)(47,175,57,165)(48,174,58,164)(49,173,59,163)(50,172,60,162)(61,261,71,271)(62,280,72,270)(63,279,73,269)(64,278,74,268)(65,277,75,267)(66,276,76,266)(67,275,77,265)(68,274,78,264)(69,273,79,263)(70,272,80,262)(81,247,91,257)(82,246,92,256)(83,245,93,255)(84,244,94,254)(85,243,95,253)(86,242,96,252)(87,241,97,251)(88,260,98,250)(89,259,99,249)(90,258,100,248)(121,155,131,145)(122,154,132,144)(123,153,133,143)(124,152,134,142)(125,151,135,141)(126,150,136,160)(127,149,137,159)(128,148,138,158)(129,147,139,157)(130,146,140,156)(201,301,211,311)(202,320,212,310)(203,319,213,309)(204,318,214,308)(205,317,215,307)(206,316,216,306)(207,315,217,305)(208,314,218,304)(209,313,219,303)(210,312,220,302)(221,282,231,292)(222,281,232,291)(223,300,233,290)(224,299,234,289)(225,298,235,288)(226,297,236,287)(227,296,237,286)(228,295,238,285)(229,294,239,284)(230,293,240,283), (1,223,163,66,113,214,145,255)(2,240,172,79,114,211,154,248)(3,237,161,72,115,208,143,241)(4,234,170,65,116,205,152,254)(5,231,179,78,117,202,141,247)(6,228,168,71,118,219,150,260)(7,225,177,64,119,216,159,253)(8,222,166,77,120,213,148,246)(9,239,175,70,101,210,157,259)(10,236,164,63,102,207,146,252)(11,233,173,76,103,204,155,245)(12,230,162,69,104,201,144,258)(13,227,171,62,105,218,153,251)(14,224,180,75,106,215,142,244)(15,221,169,68,107,212,151,257)(16,238,178,61,108,209,160,250)(17,235,167,74,109,206,149,243)(18,232,176,67,110,203,158,256)(19,229,165,80,111,220,147,249)(20,226,174,73,112,217,156,242)(21,308,121,83,183,300,49,276)(22,305,130,96,184,297,58,269)(23,302,139,89,185,294,47,262)(24,319,128,82,186,291,56,275)(25,316,137,95,187,288,45,268)(26,313,126,88,188,285,54,261)(27,310,135,81,189,282,43,274)(28,307,124,94,190,299,52,267)(29,304,133,87,191,296,41,280)(30,301,122,100,192,293,50,273)(31,318,131,93,193,290,59,266)(32,315,140,86,194,287,48,279)(33,312,129,99,195,284,57,272)(34,309,138,92,196,281,46,265)(35,306,127,85,197,298,55,278)(36,303,136,98,198,295,44,271)(37,320,125,91,199,292,53,264)(38,317,134,84,200,289,42,277)(39,314,123,97,181,286,51,270)(40,311,132,90,182,283,60,263)>;
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), (1,183,11,193)(2,182,12,192)(3,181,13,191)(4,200,14,190)(5,199,15,189)(6,198,16,188)(7,197,17,187)(8,196,18,186)(9,195,19,185)(10,194,20,184)(21,103,31,113)(22,102,32,112)(23,101,33,111)(24,120,34,110)(25,119,35,109)(26,118,36,108)(27,117,37,107)(28,116,38,106)(29,115,39,105)(30,114,40,104)(41,161,51,171)(42,180,52,170)(43,179,53,169)(44,178,54,168)(45,177,55,167)(46,176,56,166)(47,175,57,165)(48,174,58,164)(49,173,59,163)(50,172,60,162)(61,261,71,271)(62,280,72,270)(63,279,73,269)(64,278,74,268)(65,277,75,267)(66,276,76,266)(67,275,77,265)(68,274,78,264)(69,273,79,263)(70,272,80,262)(81,247,91,257)(82,246,92,256)(83,245,93,255)(84,244,94,254)(85,243,95,253)(86,242,96,252)(87,241,97,251)(88,260,98,250)(89,259,99,249)(90,258,100,248)(121,155,131,145)(122,154,132,144)(123,153,133,143)(124,152,134,142)(125,151,135,141)(126,150,136,160)(127,149,137,159)(128,148,138,158)(129,147,139,157)(130,146,140,156)(201,301,211,311)(202,320,212,310)(203,319,213,309)(204,318,214,308)(205,317,215,307)(206,316,216,306)(207,315,217,305)(208,314,218,304)(209,313,219,303)(210,312,220,302)(221,282,231,292)(222,281,232,291)(223,300,233,290)(224,299,234,289)(225,298,235,288)(226,297,236,287)(227,296,237,286)(228,295,238,285)(229,294,239,284)(230,293,240,283), (1,223,163,66,113,214,145,255)(2,240,172,79,114,211,154,248)(3,237,161,72,115,208,143,241)(4,234,170,65,116,205,152,254)(5,231,179,78,117,202,141,247)(6,228,168,71,118,219,150,260)(7,225,177,64,119,216,159,253)(8,222,166,77,120,213,148,246)(9,239,175,70,101,210,157,259)(10,236,164,63,102,207,146,252)(11,233,173,76,103,204,155,245)(12,230,162,69,104,201,144,258)(13,227,171,62,105,218,153,251)(14,224,180,75,106,215,142,244)(15,221,169,68,107,212,151,257)(16,238,178,61,108,209,160,250)(17,235,167,74,109,206,149,243)(18,232,176,67,110,203,158,256)(19,229,165,80,111,220,147,249)(20,226,174,73,112,217,156,242)(21,308,121,83,183,300,49,276)(22,305,130,96,184,297,58,269)(23,302,139,89,185,294,47,262)(24,319,128,82,186,291,56,275)(25,316,137,95,187,288,45,268)(26,313,126,88,188,285,54,261)(27,310,135,81,189,282,43,274)(28,307,124,94,190,299,52,267)(29,304,133,87,191,296,41,280)(30,301,122,100,192,293,50,273)(31,318,131,93,193,290,59,266)(32,315,140,86,194,287,48,279)(33,312,129,99,195,284,57,272)(34,309,138,92,196,281,46,265)(35,306,127,85,197,298,55,278)(36,303,136,98,198,295,44,271)(37,320,125,91,199,292,53,264)(38,317,134,84,200,289,42,277)(39,314,123,97,181,286,51,270)(40,311,132,90,182,283,60,263) );
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)], [(1,183,11,193),(2,182,12,192),(3,181,13,191),(4,200,14,190),(5,199,15,189),(6,198,16,188),(7,197,17,187),(8,196,18,186),(9,195,19,185),(10,194,20,184),(21,103,31,113),(22,102,32,112),(23,101,33,111),(24,120,34,110),(25,119,35,109),(26,118,36,108),(27,117,37,107),(28,116,38,106),(29,115,39,105),(30,114,40,104),(41,161,51,171),(42,180,52,170),(43,179,53,169),(44,178,54,168),(45,177,55,167),(46,176,56,166),(47,175,57,165),(48,174,58,164),(49,173,59,163),(50,172,60,162),(61,261,71,271),(62,280,72,270),(63,279,73,269),(64,278,74,268),(65,277,75,267),(66,276,76,266),(67,275,77,265),(68,274,78,264),(69,273,79,263),(70,272,80,262),(81,247,91,257),(82,246,92,256),(83,245,93,255),(84,244,94,254),(85,243,95,253),(86,242,96,252),(87,241,97,251),(88,260,98,250),(89,259,99,249),(90,258,100,248),(121,155,131,145),(122,154,132,144),(123,153,133,143),(124,152,134,142),(125,151,135,141),(126,150,136,160),(127,149,137,159),(128,148,138,158),(129,147,139,157),(130,146,140,156),(201,301,211,311),(202,320,212,310),(203,319,213,309),(204,318,214,308),(205,317,215,307),(206,316,216,306),(207,315,217,305),(208,314,218,304),(209,313,219,303),(210,312,220,302),(221,282,231,292),(222,281,232,291),(223,300,233,290),(224,299,234,289),(225,298,235,288),(226,297,236,287),(227,296,237,286),(228,295,238,285),(229,294,239,284),(230,293,240,283)], [(1,223,163,66,113,214,145,255),(2,240,172,79,114,211,154,248),(3,237,161,72,115,208,143,241),(4,234,170,65,116,205,152,254),(5,231,179,78,117,202,141,247),(6,228,168,71,118,219,150,260),(7,225,177,64,119,216,159,253),(8,222,166,77,120,213,148,246),(9,239,175,70,101,210,157,259),(10,236,164,63,102,207,146,252),(11,233,173,76,103,204,155,245),(12,230,162,69,104,201,144,258),(13,227,171,62,105,218,153,251),(14,224,180,75,106,215,142,244),(15,221,169,68,107,212,151,257),(16,238,178,61,108,209,160,250),(17,235,167,74,109,206,149,243),(18,232,176,67,110,203,158,256),(19,229,165,80,111,220,147,249),(20,226,174,73,112,217,156,242),(21,308,121,83,183,300,49,276),(22,305,130,96,184,297,58,269),(23,302,139,89,185,294,47,262),(24,319,128,82,186,291,56,275),(25,316,137,95,187,288,45,268),(26,313,126,88,188,285,54,261),(27,310,135,81,189,282,43,274),(28,307,124,94,190,299,52,267),(29,304,133,87,191,296,41,280),(30,301,122,100,192,293,50,273),(31,318,131,93,193,290,59,266),(32,315,140,86,194,287,48,279),(33,312,129,99,195,284,57,272),(34,309,138,92,196,281,46,265),(35,306,127,85,197,298,55,278),(36,303,136,98,198,295,44,271),(37,320,125,91,199,292,53,264),(38,317,134,84,200,289,42,277),(39,314,123,97,181,286,51,270),(40,311,132,90,182,283,60,263)])
Matrix representation ►G ⊆ GL8(𝔽41)
| 22 | 2 | 0 | 0 | 0 | 0 | 0 | 0 |
| 24 | 19 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 30 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 40 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 32 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 34 | 9 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 19 | 30 | 17 | 16 |
| 0 | 0 | 0 | 0 | 8 | 6 | 33 | 16 |
| 0 | 0 | 0 | 0 | 25 | 22 | 33 | 35 |
| 0 | 0 | 0 | 0 | 0 | 22 | 11 | 24 |
| 14 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 14 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 14 | 13 | 33 |
| 0 | 0 | 0 | 0 | 26 | 6 | 22 | 5 |
| 0 | 0 | 0 | 0 | 35 | 19 | 36 | 18 |
| 0 | 0 | 0 | 0 | 8 | 32 | 28 | 27 |
G:=sub<GL(8,GF(41))| [22,24,0,0,0,0,0,0,2,19,0,0,0,0,0,0,0,0,30,1,0,0,0,0,0,0,1,11,0,0,0,0,0,0,0,0,1,1,1,1,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0],[32,34,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,19,8,25,0,0,0,0,0,30,6,22,22,0,0,0,0,17,33,33,11,0,0,0,0,16,16,35,24],[14,0,0,0,0,0,0,0,0,14,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,13,26,35,8,0,0,0,0,14,6,19,32,0,0,0,0,13,22,36,28,0,0,0,0,33,5,18,27] >;
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | ··· | 4P | 5 | 8A | ··· | 8H | 8I | ··· | 8T | 10A | 10B | 10C | 20A | ··· | 20F |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 8 | ··· | 8 | 8 | ··· | 8 | 10 | 10 | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 4 | 5 | ··· | 5 | 10 | ··· | 10 | 4 | 4 | 4 | 8 | ··· | 8 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 |
| type | + | + | + | + | + | - | + | + | - | - | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C8 | Q8 | C4○D4 | C8○D4 | F5 | C2×F5 | D5⋊C8 | D4.F5 | Q8×F5 |
| kernel | Dic10⋊C8 | C4×C5⋊C8 | C20⋊C8 | Dic5⋊C8 | Dic5⋊3Q8 | C10.D4 | C5×C4⋊C4 | C2×Dic10 | Dic10 | C5⋊C8 | Dic5 | C10 | C4⋊C4 | C2×C4 | C4 | C2 | C2 |
| # reps | 1 | 3 | 1 | 2 | 1 | 4 | 2 | 2 | 16 | 2 | 2 | 4 | 1 | 3 | 4 | 1 | 1 |
In GAP, Magma, Sage, TeX
Dic_{10}\rtimes C_8 % in TeX
G:=Group("Dic10:C8"); // GroupNames label
G:=SmallGroup(320,1041);
// by ID
G=gap.SmallGroup(320,1041);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,120,219,268,136,6278,1595]);
// Polycyclic
G:=Group<a,b,c|a^20=c^8=1,b^2=a^10,b*a*b^-1=a^-1,c*a*c^-1=a^13,b*c=c*b>;
// generators/relations
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