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G = Dic104Q8order 320 = 26·5

2nd semidirect product of Dic10 and Q8 acting via Q8/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic104Q8, C20.14SD16, C42.42D10, C4⋊C8.10D5, C4.47(Q8×D5), C52(Q8⋊Q8), C406C4.9C2, (C2×C8).134D10, (C2×C4).138D20, (C2×C20).127D4, C4.8(C40⋊C2), C20.106(C2×Q8), (C4×C20).77C22, C202Q8.13C2, C10.13(C2×SD16), C20.290(C4○D4), (C2×C40).141C22, (C2×C20).761C23, (C4×Dic10).13C2, C20.44D4.5C2, C22.124(C2×D20), C10.34(C22⋊Q8), C4⋊Dic5.22C22, C4.114(D42D5), C2.15(D102Q8), C2.21(C8.D10), C10.18(C8.C22), (C2×Dic10).223C22, (C5×C4⋊C8).15C2, C2.16(C2×C40⋊C2), (C2×C10).144(C2×D4), (C2×C4).706(C22×D5), SmallGroup(320,478)

Series: Derived Chief Lower central Upper central

C1C2×C20 — Dic104Q8
C1C5C10C20C2×C20C2×Dic10C4×Dic10 — Dic104Q8
C5C10C2×C20 — Dic104Q8
C1C22C42C4⋊C8

Generators and relations for Dic104Q8
 G = < a,b,c,d | a20=c4=1, b2=a10, d2=c2, bab-1=cac-1=a-1, ad=da, cbc-1=a5b, bd=db, dcd-1=c-1 >

Subgroups: 374 in 96 conjugacy classes, 45 normal (29 characteristic)
C1, C2 [×3], C4 [×2], C4 [×2], C4 [×6], C22, C5, C8 [×2], C2×C4 [×3], C2×C4 [×4], Q8 [×5], C10 [×3], C42, C42, C4⋊C4 [×5], C2×C8 [×2], C2×Q8 [×2], Dic5 [×5], C20 [×2], C20 [×2], C20, C2×C10, Q8⋊C4 [×2], C4⋊C8, C4.Q8 [×2], C4×Q8, C4⋊Q8, C40 [×2], Dic10 [×2], Dic10 [×3], C2×Dic5 [×4], C2×C20 [×3], Q8⋊Q8, C4×Dic5, C10.D4, C4⋊Dic5, C4⋊Dic5 [×2], C4⋊Dic5, C4×C20, C2×C40 [×2], C2×Dic10, C2×Dic10, C20.44D4 [×2], C406C4 [×2], C5×C4⋊C8, C4×Dic10, C202Q8, Dic104Q8
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], Q8 [×2], C23, D5, SD16 [×2], C2×D4, C2×Q8, C4○D4, D10 [×3], C22⋊Q8, C2×SD16, C8.C22, D20 [×2], C22×D5, Q8⋊Q8, C40⋊C2 [×2], C2×D20, D42D5, Q8×D5, D102Q8, C2×C40⋊C2, C8.D10, Dic104Q8

Smallest permutation representation of Dic104Q8
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)
(1 195 11 185)(2 194 12 184)(3 193 13 183)(4 192 14 182)(5 191 15 181)(6 190 16 200)(7 189 17 199)(8 188 18 198)(9 187 19 197)(10 186 20 196)(21 253 31 243)(22 252 32 242)(23 251 33 241)(24 250 34 260)(25 249 35 259)(26 248 36 258)(27 247 37 257)(28 246 38 256)(29 245 39 255)(30 244 40 254)(41 138 51 128)(42 137 52 127)(43 136 53 126)(44 135 54 125)(45 134 55 124)(46 133 56 123)(47 132 57 122)(48 131 58 121)(49 130 59 140)(50 129 60 139)(61 228 71 238)(62 227 72 237)(63 226 73 236)(64 225 74 235)(65 224 75 234)(66 223 76 233)(67 222 77 232)(68 221 78 231)(69 240 79 230)(70 239 80 229)(81 149 91 159)(82 148 92 158)(83 147 93 157)(84 146 94 156)(85 145 95 155)(86 144 96 154)(87 143 97 153)(88 142 98 152)(89 141 99 151)(90 160 100 150)(101 270 111 280)(102 269 112 279)(103 268 113 278)(104 267 114 277)(105 266 115 276)(106 265 116 275)(107 264 117 274)(108 263 118 273)(109 262 119 272)(110 261 120 271)(161 295 171 285)(162 294 172 284)(163 293 173 283)(164 292 174 282)(165 291 175 281)(166 290 176 300)(167 289 177 299)(168 288 178 298)(169 287 179 297)(170 286 180 296)(201 314 211 304)(202 313 212 303)(203 312 213 302)(204 311 214 301)(205 310 215 320)(206 309 216 319)(207 308 217 318)(208 307 218 317)(209 306 219 316)(210 305 220 315)
(1 130 271 36)(2 129 272 35)(3 128 273 34)(4 127 274 33)(5 126 275 32)(6 125 276 31)(7 124 277 30)(8 123 278 29)(9 122 279 28)(10 121 280 27)(11 140 261 26)(12 139 262 25)(13 138 263 24)(14 137 264 23)(15 136 265 22)(16 135 266 21)(17 134 267 40)(18 133 268 39)(19 132 269 38)(20 131 270 37)(41 103 260 188)(42 102 241 187)(43 101 242 186)(44 120 243 185)(45 119 244 184)(46 118 245 183)(47 117 246 182)(48 116 247 181)(49 115 248 200)(50 114 249 199)(51 113 250 198)(52 112 251 197)(53 111 252 196)(54 110 253 195)(55 109 254 194)(56 108 255 193)(57 107 256 192)(58 106 257 191)(59 105 258 190)(60 104 259 189)(61 292 96 312)(62 291 97 311)(63 290 98 310)(64 289 99 309)(65 288 100 308)(66 287 81 307)(67 286 82 306)(68 285 83 305)(69 284 84 304)(70 283 85 303)(71 282 86 302)(72 281 87 301)(73 300 88 320)(74 299 89 319)(75 298 90 318)(76 297 91 317)(77 296 92 316)(78 295 93 315)(79 294 94 314)(80 293 95 313)(141 201 235 162)(142 220 236 161)(143 219 237 180)(144 218 238 179)(145 217 239 178)(146 216 240 177)(147 215 221 176)(148 214 222 175)(149 213 223 174)(150 212 224 173)(151 211 225 172)(152 210 226 171)(153 209 227 170)(154 208 228 169)(155 207 229 168)(156 206 230 167)(157 205 231 166)(158 204 232 165)(159 203 233 164)(160 202 234 163)
(1 144 271 238)(2 145 272 239)(3 146 273 240)(4 147 274 221)(5 148 275 222)(6 149 276 223)(7 150 277 224)(8 151 278 225)(9 152 279 226)(10 153 280 227)(11 154 261 228)(12 155 262 229)(13 156 263 230)(14 157 264 231)(15 158 265 232)(16 159 266 233)(17 160 267 234)(18 141 268 235)(19 142 269 236)(20 143 270 237)(21 203 135 164)(22 204 136 165)(23 205 137 166)(24 206 138 167)(25 207 139 168)(26 208 140 169)(27 209 121 170)(28 210 122 171)(29 211 123 172)(30 212 124 173)(31 213 125 174)(32 214 126 175)(33 215 127 176)(34 216 128 177)(35 217 129 178)(36 218 130 179)(37 219 131 180)(38 220 132 161)(39 201 133 162)(40 202 134 163)(41 299 260 319)(42 300 241 320)(43 281 242 301)(44 282 243 302)(45 283 244 303)(46 284 245 304)(47 285 246 305)(48 286 247 306)(49 287 248 307)(50 288 249 308)(51 289 250 309)(52 290 251 310)(53 291 252 311)(54 292 253 312)(55 293 254 313)(56 294 255 314)(57 295 256 315)(58 296 257 316)(59 297 258 317)(60 298 259 318)(61 195 96 110)(62 196 97 111)(63 197 98 112)(64 198 99 113)(65 199 100 114)(66 200 81 115)(67 181 82 116)(68 182 83 117)(69 183 84 118)(70 184 85 119)(71 185 86 120)(72 186 87 101)(73 187 88 102)(74 188 89 103)(75 189 90 104)(76 190 91 105)(77 191 92 106)(78 192 93 107)(79 193 94 108)(80 194 95 109)

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

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

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

59 conjugacy classes

class 1 2A2B2C4A4B4C4D4E4F4G4H4I4J4K5A5B8A8B8C8D10A···10F20A···20H20I···20P40A···40P
order12224444444444455888810···1020···2020···2040···40
size1111222242020202040402244442···22···24···44···4

59 irreducible representations

dim1111112222222224444
type++++++-+++++----
imageC1C2C2C2C2C2Q8D4D5SD16C4○D4D10D10D20C40⋊C2C8.C22D42D5Q8×D5C8.D10
kernelDic104Q8C20.44D4C406C4C5×C4⋊C8C4×Dic10C202Q8Dic10C2×C20C4⋊C8C20C20C42C2×C8C2×C4C4C10C4C4C2
# reps12211122242248161224

Matrix representation of Dic104Q8 in GL6(𝔽41)

1400000
8340000
0020900
0012100
0000400
0000040
,
2180000
27200000
00391200
003200
0000132
00003928
,
3880000
4030000
001000
00324000
00003928
0000132
,
4000000
0400000
001000
000100
0000623
00001835

G:=sub<GL(6,GF(41))| [1,8,0,0,0,0,40,34,0,0,0,0,0,0,20,1,0,0,0,0,9,21,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[21,27,0,0,0,0,8,20,0,0,0,0,0,0,39,3,0,0,0,0,12,2,0,0,0,0,0,0,13,39,0,0,0,0,2,28],[38,40,0,0,0,0,8,3,0,0,0,0,0,0,1,32,0,0,0,0,0,40,0,0,0,0,0,0,39,13,0,0,0,0,28,2],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,6,18,0,0,0,0,23,35] >;

Dic104Q8 in GAP, Magma, Sage, TeX

{\rm Dic}_{10}\rtimes_4Q_8
% in TeX

G:=Group("Dic10:4Q8");
// GroupNames label

G:=SmallGroup(320,478);
// by ID

G=gap.SmallGroup(320,478);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,120,254,219,142,1123,136,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^20=c^4=1,b^2=a^10,d^2=c^2,b*a*b^-1=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=a^5*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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