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G = (C2×Dic5)⋊6Q8  order 320 = 26·5

2nd semidirect product of C2×Dic5 and Q8 acting via Q8/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C2×Dic5)⋊6Q8, C10.30(C4×Q8), (C2×C4).141D20, (C2×C20).137D4, C10.25(C4⋊Q8), (C2×Dic10)⋊21C4, C22.20(Q8×D5), C22.45(C2×D20), C4.9(D10⋊C4), C20.69(C22⋊C4), C2.3(D10⋊2Q8), (C22×C4).332D10, C10.44(C22⋊Q8), C2.2(Dic5⋊Q8), C10.39(C4.4D4), C2.2(C20.17D4), C23.288(C22×D5), C2.11(Dic5⋊3Q8), C22.53(D4⋊2D5), (C22×C20).141C22, (C22×C10).338C23, C5⋊4(C23.67C23), (C22×Dic10).12C2, C10.10C42.16C2, (C22×Dic5).50C22, (C2×C4⋊C4).12D5, (C2×C4).78(C4×D5), (C10×C4⋊C4).10C2, (C2×C4×Dic5).4C2, (C2×C10).74(C2×Q8), C22.132(C2×C4×D5), (C2×C20).254(C2×C4), (C2×C10).150(C2×D4), C10.82(C2×C22⋊C4), C22.62(C2×C5⋊D4), C2.14(C2×D10⋊C4), (C2×C4).126(C5⋊D4), (C2×Dic5).31(C2×C4), (C2×C10).150(C4○D4), (C2×C10).215(C22×C4), SmallGroup(320,601)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C10 — (C2×Dic5)⋊6Q8
Chief series
C1 — C5 — C10 — C2×C10 — C22×C10 — C22×Dic5 — C2×C4×Dic5 — (C2×Dic5)⋊6Q8
Lower central
C5 — C2×C10 — (C2×Dic5)⋊6Q8
Upper central
C1 — C23 — C2×C4⋊C4

Generators and relations for (C2×Dic5)⋊6Q8
 G = < a,b,c,d,e | a2=b10=d4=1, c2=b5, e2=d2, ab=ba, dcd-1=ac=ca, ad=da, ae=ea, cbc-1=dbd-1=b-1, be=eb, ce=ec, ede-1=d-1 >

Subgroups: 606 in 186 conjugacy classes, 83 normal (23 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, Q8, C23, C10, C10, C42, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×Q8, Dic5, C20, C20, C2×C10, C2×C10, C2.C42, C2×C42, C2×C4⋊C4, C22×Q8, Dic10, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×C10, C23.67C23, C4×Dic5, C5×C4⋊C4, C2×Dic10, C2×Dic10, C22×Dic5, C22×C20, C22×C20, C10.10C42, C2×C4×Dic5, C10×C4⋊C4, C22×Dic10, (C2×Dic5)⋊6Q8
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D5, C22⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, D10, C2×C22⋊C4, C4×Q8, C22⋊Q8, C4.4D4, C4⋊Q8, C4×D5, D20, C5⋊D4, C22×D5, C23.67C23, D10⋊C4, C2×C4×D5, C2×D20, D4⋊2D5, Q8×D5, C2×C5⋊D4, Dic5⋊3Q8, D10⋊2Q8, C2×D10⋊C4, C20.17D4, Dic5⋊Q8, (C2×Dic5)⋊6Q8

Smallest permutation representation of (C2×Dic5)⋊6Q8
►Regular action on 320 points
Generators in S320
(1 21)(2 22)(3 23)(4 24)(5 25)(6 26)(7 27)(8 28)(9 29)(10 30)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)(41 61)(42 62)(43 63)(44 64)(45 65)(46 66)(47 67)(48 68)(49 69)(50 70)(51 71)(52 72)(53 73)(54 74)(55 75)(56 76)(57 77)(58 78)(59 79)(60 80)(81 101)(82 102)(83 103)(84 104)(85 105)(86 106)(87 107)(88 108)(89 109)(90 110)(91 111)(92 112)(93 113)(94 114)(95 115)(96 116)(97 117)(98 118)(99 119)(100 120)(121 141)(122 142)(123 143)(124 144)(125 145)(126 146)(127 147)(128 148)(129 149)(130 150)(131 151)(132 152)(133 153)(134 154)(135 155)(136 156)(137 157)(138 158)(139 159)(140 160)(161 181)(162 182)(163 183)(164 184)(165 185)(166 186)(167 187)(168 188)(169 189)(170 190)(171 191)(172 192)(173 193)(174 194)(175 195)(176 196)(177 197)(178 198)(179 199)(180 200)(201 221)(202 222)(203 223)(204 224)(205 225)(206 226)(207 227)(208 228)(209 229)(210 230)(211 231)(212 232)(213 233)(214 234)(215 235)(216 236)(217 237)(218 238)(219 239)(220 240)(241 261)(242 262)(243 263)(244 264)(245 265)(246 266)(247 267)(248 268)(249 269)(250 270)(251 271)(252 272)(253 273)(254 274)(255 275)(256 276)(257 277)(258 278)(259 279)(260 280)(281 301)(282 302)(283 303)(284 304)(285 305)(286 306)(287 307)(288 308)(289 309)(290 310)(291 311)(292 312)(293 313)(294 314)(295 315)(296 316)(297 317)(298 318)(299 319)(300 320)

(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)

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

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

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

68 conjugacy classes

class 1 2A···2G4A4B4C4D4E4F4G4H4I···4P4Q4R4S4T5A5B10A···10N20A···20X
order12···2444444444···444445510···1020···20
size11···12222444410···1020202020222···24···4

68 irreducible representations

dim1111112222222244
type+++++-++++--
imageC1C2C2C2C2C4Q8D4D5C4○D4D10C4×D5D20C5⋊D4D4⋊2D5Q8×D5
kernel(C2×Dic5)⋊6Q8C10.10C42C2×C4×Dic5C10×C4⋊C4C22×Dic10C2×Dic10C2×Dic5C2×C20C2×C4⋊C4C2×C10C22×C4C2×C4C2×C4C2×C4C22C22
# reps1411184424688844

Matrix representation of (C2×Dic5)⋊6Q8 ►in GL6(𝔽41)

4000000
0400000
0040000
0004000
0000400
0000040
,
6400000
100000
0035100
0040000
000010
000001
,
2130000
25390000
00263900
00311500
0000400
0000391
,
100000
6400000
005300
00333600
0000140
0000240
,
4000000
0400000
001000
000100
000090
00001832

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[6,1,0,0,0,0,40,0,0,0,0,0,0,0,35,40,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[2,25,0,0,0,0,13,39,0,0,0,0,0,0,26,31,0,0,0,0,39,15,0,0,0,0,0,0,40,39,0,0,0,0,0,1],[1,6,0,0,0,0,0,40,0,0,0,0,0,0,5,33,0,0,0,0,3,36,0,0,0,0,0,0,1,2,0,0,0,0,40,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,18,0,0,0,0,0,32] >;
 

(C2×Dic5)⋊6Q8 in GAP, Magma, Sage, TeX 

(C_2\times {\rm Dic}_5)\rtimes_6Q_8
 
% in TeX
 
G:=Group("(C2xDic5):6Q8");
 
// GroupNames label
 
G:=SmallGroup(320,601);
 
// by ID
 
G=gap.SmallGroup(320,601);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,120,254,219,310,12550]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^10=d^4=1,c^2=b^5,e^2=d^2,a*b=b*a,d*c*d^-1=a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=d*b*d^-1=b^-1,b*e=e*b,c*e=e*c,e*d*e^-1=d^-1>;
 
// generators/relations
 

׿
×
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