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G = Dic10⋊7Q8  order 320 = 26·5

5th semidirect product of Dic10 and Q8 acting via Q8/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic10⋊7Q8, C42.146D10, C10.942- (1+4), C4.15(Q8×D5), C5⋊4(Q8⋊3Q8), C20⋊Q8.13C2, C20.47(C2×Q8), C4⋊C4.202D10, C42.C2.6D5, (C2×C20).84C23, Dic5.25(C2×Q8), C10.39(C22×Q8), (C2×C10).229C24, (C4×C20).190C22, (C4×Dic10).24C2, C4.Dic10.13C2, Dic5.66(C4○D4), Dic5⋊3Q8.11C2, C4⋊Dic5.378C22, Dic5.Q8.2C2, C22.250(C23×D5), (C4×Dic5).145C22, (C2×Dic5).119C23, C2.55(D4.10D10), (C2×Dic10).306C22, C10.D4.143C22, C2.22(C2×Q8×D5), C2.82(D5×C4○D4), C10.193(C2×C4○D4), (C2×C4).75(C22×D5), (C5×C42.C2).5C2, (C5×C4⋊C4).184C22, SmallGroup(320,1357)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C10 — Dic10⋊7Q8
Chief series
C1 — C5 — C10 — C2×C10 — C2×Dic5 — C4×Dic5 — Dic5⋊3Q8 — Dic10⋊7Q8
Lower central
C5 — C2×C10 — Dic10⋊7Q8

Subgroups: 566 in 200 conjugacy classes, 105 normal (43 characteristic)
C1, C2 [×3], C4 [×2], C4 [×17], C22, C5, C2×C4 [×3], C2×C4 [×4], C2×C4 [×8], Q8 [×10], C10 [×3], C42, C42 [×8], C4⋊C4 [×2], C4⋊C4 [×4], C4⋊C4 [×16], C2×Q8 [×4], Dic5 [×6], Dic5 [×5], C20 [×2], C20 [×6], C2×C10, C4×Q8 [×6], C42.C2, C42.C2 [×5], C4⋊Q8 [×3], Dic10 [×4], Dic10 [×6], C2×Dic5 [×4], C2×Dic5 [×4], C2×C20 [×3], C2×C20 [×4], Q8⋊3Q8, C4×Dic5 [×2], C4×Dic5 [×6], C10.D4 [×2], C10.D4 [×10], C4⋊Dic5 [×2], C4⋊Dic5 [×2], C4×C20, C5×C4⋊C4 [×2], C5×C4⋊C4 [×4], C2×Dic10 [×2], C2×Dic10 [×2], C4×Dic10 [×2], Dic5⋊3Q8 [×2], Dic5⋊3Q8 [×2], C20⋊Q8, C20⋊Q8 [×2], Dic5.Q8 [×4], C4.Dic10, C5×C42.C2, Dic10⋊7Q8

Quotients:
C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], D5, C2×Q8 [×6], C4○D4 [×2], C24, D10 [×7], C22×Q8, C2×C4○D4, 2- (1+4), C22×D5 [×7], Q8⋊3Q8, Q8×D5 [×2], C23×D5, C2×Q8×D5, D5×C4○D4, D4.10D10, Dic10⋊7Q8

Generators and relations
 G = < a,b,c,d | a20=c4=1, b2=a10, d2=c2, bab-1=a-1, ac=ca, dad-1=a9, cbc-1=dbd-1=a10b, dcd-1=c-1 >

Smallest permutation representation
►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)

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

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

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

Matrix representation ►G ⊆ GL6(𝔽41)

0320000
3200000
0031000
0031400
000010
000001
,
39280000
1320000
00383700
002300
000010
000001
,
0400000
4000000
001000
000100
000001
0000400
,
010000
100000
00383700
002300
0000235
00003539

G:=sub<GL(6,GF(41))| [0,32,0,0,0,0,32,0,0,0,0,0,0,0,31,31,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[39,13,0,0,0,0,28,2,0,0,0,0,0,0,38,2,0,0,0,0,37,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,40,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,1,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,38,2,0,0,0,0,37,3,0,0,0,0,0,0,2,35,0,0,0,0,35,39] >;
 

53 conjugacy classes

class 1 2A2B2C4A4B4C4D4E···4I4J···4Q4R4S4T4U5A5B10A···10F20A···20L20M···20T
order122244444···44···444445510···1020···2020···20
size111122224···410···1020202020222···24···48···8

53 irreducible representations

dim1111111222224444
type+++++++-+++---
imageC1C2C2C2C2C2C2Q8D5C4○D4D10D102- (1+4)Q8×D5D5×C4○D4D4.10D10
kernelDic10⋊7Q8C4×Dic10Dic5⋊3Q8C20⋊Q8Dic5.Q8C4.Dic10C5×C42.C2Dic10C42.C2Dic5C42C4⋊C4C10C4C2C2
# reps12434114242121444

In GAP, Magma, Sage, TeX 

Dic_{10}\rtimes_7Q_8
 
% in TeX
 
G:=Group("Dic10:7Q8");
 
// GroupNames label
 
G:=SmallGroup(320,1357);
 
// by ID
 
G=gap.SmallGroup(320,1357);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,120,219,268,1571,297,192,12550]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^20=c^4=1,b^2=a^10,d^2=c^2,b*a*b^-1=a^-1,a*c=c*a,d*a*d^-1=a^9,c*b*c^-1=d*b*d^-1=a^10*b,d*c*d^-1=c^-1>;
 
// generators/relations
 

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