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G = Dic1010Q8order 320 = 26·5

The semidirect product of Dic10 and Q8 acting through Inn(Dic10)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic1010Q8, C42.121D10, C10.642- 1+4, C4.49(Q8×D5), C51(Q83Q8), (C4×Q8).10D5, C4⋊C4.321D10, (Q8×C20).11C2, C20.107(C2×Q8), C4.17(C4○D20), (C2×Q8).175D10, C202Q8.25C2, Dic5.22(C2×Q8), C20.115(C4○D4), C10.27(C22×Q8), (C2×C20).589C23, (C2×C10).111C24, (C4×C20).164C22, Dic5⋊Q8.9C2, (C4×Dic10).21C2, C4⋊Dic5.42C22, C20.6Q8.10C2, Dic53Q8.10C2, Dic5.Q8.1C2, (Q8×C10).211C22, (C4×Dic5).88C22, (C2×Dic5).50C23, C22.136(C23×D5), C2.21(D4.10D10), (C2×Dic10).152C22, C10.D4.114C22, C2.11(C2×Q8×D5), C10.52(C2×C4○D4), C2.59(C2×C4○D20), (C5×C4⋊C4).339C22, (C2×C4).166(C22×D5), SmallGroup(320,1239)

Series: Derived Chief Lower central Upper central

C1C2×C10 — Dic1010Q8
C1C5C10C2×C10C2×Dic5C2×Dic10C4×Dic10 — Dic1010Q8
C5C2×C10 — Dic1010Q8
C1C22C4×Q8

Generators and relations for Dic1010Q8
 G = < a,b,c,d | a20=c4=1, b2=a10, d2=c2, bab-1=a-1, ac=ca, ad=da, cbc-1=a10b, bd=db, dcd-1=c-1 >

Subgroups: 550 in 200 conjugacy classes, 107 normal (29 characteristic)
C1, C2 [×3], C4 [×4], C4 [×15], C22, C5, C2×C4 [×3], C2×C4 [×4], C2×C4 [×8], Q8 [×10], C10 [×3], C42, C42 [×2], C42 [×6], C4⋊C4, C4⋊C4 [×2], C4⋊C4 [×19], C2×Q8, C2×Q8 [×3], Dic5 [×4], Dic5 [×6], C20 [×4], C20 [×5], C2×C10, C4×Q8, C4×Q8 [×5], C42.C2 [×6], C4⋊Q8 [×3], Dic10 [×4], Dic10 [×4], C2×Dic5 [×8], C2×C20 [×3], C2×C20 [×4], C5×Q8 [×2], Q83Q8, C4×Dic5 [×6], C10.D4 [×14], C4⋊Dic5, C4⋊Dic5 [×4], C4×C20, C4×C20 [×2], C5×C4⋊C4, C5×C4⋊C4 [×2], C2×Dic10, C2×Dic10 [×2], Q8×C10, C4×Dic10, C4×Dic10 [×2], C202Q8, C20.6Q8 [×2], Dic53Q8 [×2], Dic5.Q8 [×4], Dic5⋊Q8 [×2], Q8×C20, Dic1010Q8
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], Q83Q8, C4○D20 [×2], Q8×D5 [×2], C23×D5, C2×C4○D20, C2×Q8×D5, D4.10D10, Dic1010Q8

Smallest permutation representation of Dic1010Q8
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)
(1 301 11 311)(2 320 12 310)(3 319 13 309)(4 318 14 308)(5 317 15 307)(6 316 16 306)(7 315 17 305)(8 314 18 304)(9 313 19 303)(10 312 20 302)(21 71 31 61)(22 70 32 80)(23 69 33 79)(24 68 34 78)(25 67 35 77)(26 66 36 76)(27 65 37 75)(28 64 38 74)(29 63 39 73)(30 62 40 72)(41 268 51 278)(42 267 52 277)(43 266 53 276)(44 265 54 275)(45 264 55 274)(46 263 56 273)(47 262 57 272)(48 261 58 271)(49 280 59 270)(50 279 60 269)(81 300 91 290)(82 299 92 289)(83 298 93 288)(84 297 94 287)(85 296 95 286)(86 295 96 285)(87 294 97 284)(88 293 98 283)(89 292 99 282)(90 291 100 281)(101 170 111 180)(102 169 112 179)(103 168 113 178)(104 167 114 177)(105 166 115 176)(106 165 116 175)(107 164 117 174)(108 163 118 173)(109 162 119 172)(110 161 120 171)(121 212 131 202)(122 211 132 201)(123 210 133 220)(124 209 134 219)(125 208 135 218)(126 207 136 217)(127 206 137 216)(128 205 138 215)(129 204 139 214)(130 203 140 213)(141 246 151 256)(142 245 152 255)(143 244 153 254)(144 243 154 253)(145 242 155 252)(146 241 156 251)(147 260 157 250)(148 259 158 249)(149 258 159 248)(150 257 160 247)(181 226 191 236)(182 225 192 235)(183 224 193 234)(184 223 194 233)(185 222 195 232)(186 221 196 231)(187 240 197 230)(188 239 198 229)(189 238 199 228)(190 237 200 227)
(1 226 126 31)(2 227 127 32)(3 228 128 33)(4 229 129 34)(5 230 130 35)(6 231 131 36)(7 232 132 37)(8 233 133 38)(9 234 134 39)(10 235 135 40)(11 236 136 21)(12 237 137 22)(13 238 138 23)(14 239 139 24)(15 240 140 25)(16 221 121 26)(17 222 122 27)(18 223 123 28)(19 224 124 29)(20 225 125 30)(41 177 293 151)(42 178 294 152)(43 179 295 153)(44 180 296 154)(45 161 297 155)(46 162 298 156)(47 163 299 157)(48 164 300 158)(49 165 281 159)(50 166 282 160)(51 167 283 141)(52 168 284 142)(53 169 285 143)(54 170 286 144)(55 171 287 145)(56 172 288 146)(57 173 289 147)(58 174 290 148)(59 175 291 149)(60 176 292 150)(61 311 191 217)(62 312 192 218)(63 313 193 219)(64 314 194 220)(65 315 195 201)(66 316 196 202)(67 317 197 203)(68 318 198 204)(69 319 199 205)(70 320 200 206)(71 301 181 207)(72 302 182 208)(73 303 183 209)(74 304 184 210)(75 305 185 211)(76 306 186 212)(77 307 187 213)(78 308 188 214)(79 309 189 215)(80 310 190 216)(81 249 271 117)(82 250 272 118)(83 251 273 119)(84 252 274 120)(85 253 275 101)(86 254 276 102)(87 255 277 103)(88 256 278 104)(89 257 279 105)(90 258 280 106)(91 259 261 107)(92 260 262 108)(93 241 263 109)(94 242 264 110)(95 243 265 111)(96 244 266 112)(97 245 267 113)(98 246 268 114)(99 247 269 115)(100 248 270 116)
(1 288 126 56)(2 289 127 57)(3 290 128 58)(4 291 129 59)(5 292 130 60)(6 293 131 41)(7 294 132 42)(8 295 133 43)(9 296 134 44)(10 297 135 45)(11 298 136 46)(12 299 137 47)(13 300 138 48)(14 281 139 49)(15 282 140 50)(16 283 121 51)(17 284 122 52)(18 285 123 53)(19 286 124 54)(20 287 125 55)(21 156 236 162)(22 157 237 163)(23 158 238 164)(24 159 239 165)(25 160 240 166)(26 141 221 167)(27 142 222 168)(28 143 223 169)(29 144 224 170)(30 145 225 171)(31 146 226 172)(32 147 227 173)(33 148 228 174)(34 149 229 175)(35 150 230 176)(36 151 231 177)(37 152 232 178)(38 153 233 179)(39 154 234 180)(40 155 235 161)(61 241 191 109)(62 242 192 110)(63 243 193 111)(64 244 194 112)(65 245 195 113)(66 246 196 114)(67 247 197 115)(68 248 198 116)(69 249 199 117)(70 250 200 118)(71 251 181 119)(72 252 182 120)(73 253 183 101)(74 254 184 102)(75 255 185 103)(76 256 186 104)(77 257 187 105)(78 258 188 106)(79 259 189 107)(80 260 190 108)(81 205 271 319)(82 206 272 320)(83 207 273 301)(84 208 274 302)(85 209 275 303)(86 210 276 304)(87 211 277 305)(88 212 278 306)(89 213 279 307)(90 214 280 308)(91 215 261 309)(92 216 262 310)(93 217 263 311)(94 218 264 312)(95 219 265 313)(96 220 266 314)(97 201 267 315)(98 202 268 316)(99 203 269 317)(100 204 270 318)

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

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

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

65 conjugacy classes

class 1 2A2B2C4A···4H4I4J4K4L4M4N4O4P···4U5A5B10A···10F20A···20H20I···20AF
order12224···444444444···45510···1020···2020···20
size11112···24441010101020···20222···22···24···4

65 irreducible representations

dim111111112222222444
type++++++++-++++---
imageC1C2C2C2C2C2C2C2Q8D5C4○D4D10D10D10C4○D202- 1+4Q8×D5D4.10D10
kernelDic1010Q8C4×Dic10C202Q8C20.6Q8Dic53Q8Dic5.Q8Dic5⋊Q8Q8×C20Dic10C4×Q8C20C42C4⋊C4C2×Q8C4C10C4C2
# reps1312242142466216144

Matrix representation of Dic1010Q8 in GL4(𝔽41) generated by

25200
391300
0010
0001
,
233500
201800
0010
0001
,
18600
352300
003223
0009
,
40000
04000
00713
003434
G:=sub<GL(4,GF(41))| [25,39,0,0,2,13,0,0,0,0,1,0,0,0,0,1],[23,20,0,0,35,18,0,0,0,0,1,0,0,0,0,1],[18,35,0,0,6,23,0,0,0,0,32,0,0,0,23,9],[40,0,0,0,0,40,0,0,0,0,7,34,0,0,13,34] >;

Dic1010Q8 in GAP, Magma, Sage, TeX

{\rm Dic}_{10}\rtimes_{10}Q_8
% in TeX

G:=Group("Dic10:10Q8");
// GroupNames label

G:=SmallGroup(320,1239);
// by ID

G=gap.SmallGroup(320,1239);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,477,232,100,185,192,12550]);
// 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,a*d=d*a,c*b*c^-1=a^10*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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