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G = Dic106Q8order 320 = 26·5

4th semidirect product of Dic10 and Q8 acting via Q8/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic106Q8, C20.21SD16, C42.85D10, C4⋊Q8.13D5, C4.14(Q8×D5), C4⋊C4.88D10, C55(Q8⋊Q8), C20.41(C2×Q8), (C2×C20).163D4, C4.8(D4.D5), C203C8.23C2, C20.88(C4○D4), C10.62(C2×SD16), (C2×C20).412C23, (C4×C20).141C22, C4.37(Q82D5), (C4×Dic10).19C2, C10.Q16.16C2, C10.78(C22⋊Q8), C20.Q8.18C2, C2.15(D103Q8), C4⋊Dic5.351C22, C2.21(C20.C23), C10.100(C8.C22), (C2×Dic10).285C22, (C5×C4⋊Q8).13C2, C2.16(C2×D4.D5), (C2×C10).543(C2×D4), (C2×C4).192(C5⋊D4), (C5×C4⋊C4).135C22, (C2×C4).509(C22×D5), C22.215(C2×C5⋊D4), (C2×C52C8).142C22, SmallGroup(320,721)

Series: Derived Chief Lower central Upper central

C1C2×C20 — Dic106Q8
C1C5C10C20C2×C20C2×Dic10C4×Dic10 — Dic106Q8
C5C10C2×C20 — Dic106Q8
C1C22C42C4⋊Q8

Generators and relations for Dic106Q8
 G = < a,b,c,d | a20=c4=1, b2=a10, d2=c2, bab-1=a-1, cac-1=a11, ad=da, cbc-1=a5b, bd=db, dcd-1=c-1 >

Subgroups: 294 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 [×2], C4⋊C4 [×3], C2×C8 [×2], C2×Q8 [×2], Dic5 [×3], C20 [×2], C20 [×2], C20 [×3], C2×C10, Q8⋊C4 [×2], C4⋊C8, C4.Q8 [×2], C4×Q8, C4⋊Q8, C52C8 [×2], Dic10 [×2], Dic10, C2×Dic5 [×2], C2×C20 [×3], C2×C20 [×2], C5×Q8 [×2], Q8⋊Q8, C2×C52C8 [×2], C4×Dic5, C10.D4, C4⋊Dic5, C4×C20, C5×C4⋊C4 [×2], C5×C4⋊C4, C2×Dic10, Q8×C10, C203C8, C20.Q8 [×2], C10.Q16 [×2], C4×Dic10, C5×C4⋊Q8, Dic106Q8
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, C5⋊D4 [×2], C22×D5, Q8⋊Q8, D4.D5 [×2], Q8×D5, Q82D5, C2×C5⋊D4, C2×D4.D5, C20.C23, D103Q8, Dic106Q8

Smallest permutation representation of Dic106Q8
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)
(1 136 11 126)(2 135 12 125)(3 134 13 124)(4 133 14 123)(5 132 15 122)(6 131 16 121)(7 130 17 140)(8 129 18 139)(9 128 19 138)(10 127 20 137)(21 312 31 302)(22 311 32 301)(23 310 33 320)(24 309 34 319)(25 308 35 318)(26 307 36 317)(27 306 37 316)(28 305 38 315)(29 304 39 314)(30 303 40 313)(41 279 51 269)(42 278 52 268)(43 277 53 267)(44 276 54 266)(45 275 55 265)(46 274 56 264)(47 273 57 263)(48 272 58 262)(49 271 59 261)(50 270 60 280)(61 249 71 259)(62 248 72 258)(63 247 73 257)(64 246 74 256)(65 245 75 255)(66 244 76 254)(67 243 77 253)(68 242 78 252)(69 241 79 251)(70 260 80 250)(81 114 91 104)(82 113 92 103)(83 112 93 102)(84 111 94 101)(85 110 95 120)(86 109 96 119)(87 108 97 118)(88 107 98 117)(89 106 99 116)(90 105 100 115)(141 282 151 292)(142 281 152 291)(143 300 153 290)(144 299 154 289)(145 298 155 288)(146 297 156 287)(147 296 157 286)(148 295 158 285)(149 294 159 284)(150 293 160 283)(161 207 171 217)(162 206 172 216)(163 205 173 215)(164 204 174 214)(165 203 175 213)(166 202 176 212)(167 201 177 211)(168 220 178 210)(169 219 179 209)(170 218 180 208)(181 223 191 233)(182 222 192 232)(183 221 193 231)(184 240 194 230)(185 239 195 229)(186 238 196 228)(187 237 197 227)(188 236 198 226)(189 235 199 225)(190 234 200 224)
(1 75 217 224)(2 66 218 235)(3 77 219 226)(4 68 220 237)(5 79 201 228)(6 70 202 239)(7 61 203 230)(8 72 204 221)(9 63 205 232)(10 74 206 223)(11 65 207 234)(12 76 208 225)(13 67 209 236)(14 78 210 227)(15 69 211 238)(16 80 212 229)(17 71 213 240)(18 62 214 231)(19 73 215 222)(20 64 216 233)(21 104 268 152)(22 115 269 143)(23 106 270 154)(24 117 271 145)(25 108 272 156)(26 119 273 147)(27 110 274 158)(28 101 275 149)(29 112 276 160)(30 103 277 151)(31 114 278 142)(32 105 279 153)(33 116 280 144)(34 107 261 155)(35 118 262 146)(36 109 263 157)(37 120 264 148)(38 111 265 159)(39 102 266 150)(40 113 267 141)(41 295 311 85)(42 286 312 96)(43 297 313 87)(44 288 314 98)(45 299 315 89)(46 290 316 100)(47 281 317 91)(48 292 318 82)(49 283 319 93)(50 294 320 84)(51 285 301 95)(52 296 302 86)(53 287 303 97)(54 298 304 88)(55 289 305 99)(56 300 306 90)(57 291 307 81)(58 282 308 92)(59 293 309 83)(60 284 310 94)(121 245 166 200)(122 256 167 191)(123 247 168 182)(124 258 169 193)(125 249 170 184)(126 260 171 195)(127 251 172 186)(128 242 173 197)(129 253 174 188)(130 244 175 199)(131 255 176 190)(132 246 177 181)(133 257 178 192)(134 248 179 183)(135 259 180 194)(136 250 161 185)(137 241 162 196)(138 252 163 187)(139 243 164 198)(140 254 165 189)
(1 298 217 88)(2 299 218 89)(3 300 219 90)(4 281 220 91)(5 282 201 92)(6 283 202 93)(7 284 203 94)(8 285 204 95)(9 286 205 96)(10 287 206 97)(11 288 207 98)(12 289 208 99)(13 290 209 100)(14 291 210 81)(15 292 211 82)(16 293 212 83)(17 294 213 84)(18 295 214 85)(19 296 215 86)(20 297 216 87)(21 257 268 192)(22 258 269 193)(23 259 270 194)(24 260 271 195)(25 241 272 196)(26 242 273 197)(27 243 274 198)(28 244 275 199)(29 245 276 200)(30 246 277 181)(31 247 278 182)(32 248 279 183)(33 249 280 184)(34 250 261 185)(35 251 262 186)(36 252 263 187)(37 253 264 188)(38 254 265 189)(39 255 266 190)(40 256 267 191)(41 231 311 62)(42 232 312 63)(43 233 313 64)(44 234 314 65)(45 235 315 66)(46 236 316 67)(47 237 317 68)(48 238 318 69)(49 239 319 70)(50 240 320 71)(51 221 301 72)(52 222 302 73)(53 223 303 74)(54 224 304 75)(55 225 305 76)(56 226 306 77)(57 227 307 78)(58 228 308 79)(59 229 309 80)(60 230 310 61)(101 130 149 175)(102 131 150 176)(103 132 151 177)(104 133 152 178)(105 134 153 179)(106 135 154 180)(107 136 155 161)(108 137 156 162)(109 138 157 163)(110 139 158 164)(111 140 159 165)(112 121 160 166)(113 122 141 167)(114 123 142 168)(115 124 143 169)(116 125 144 170)(117 126 145 171)(118 127 146 172)(119 128 147 173)(120 129 148 174)

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

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

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

47 conjugacy classes

class 1 2A2B2C4A4B4C4D4E4F4G4H4I4J4K5A5B8A8B8C8D10A···10F20A···20L20M···20T
order12224444444444455888810···1020···2020···20
size111122224882020202022202020202···24···48···8

47 irreducible representations

dim1111112222222244444
type++++++-++++---+
imageC1C2C2C2C2C2Q8D4D5SD16C4○D4D10D10C5⋊D4C8.C22D4.D5Q8×D5Q82D5C20.C23
kernelDic106Q8C203C8C20.Q8C10.Q16C4×Dic10C5×C4⋊Q8Dic10C2×C20C4⋊Q8C20C20C42C4⋊C4C2×C4C10C4C4C4C2
# reps1122112224224814224

Matrix representation of Dic106Q8 in GL6(𝔽41)

40320000
2310000
000100
00403400
0000400
0000040
,
900000
39320000
00323000
0011900
000001
000010
,
0260000
3000000
001000
000100
000001
0000400
,
100000
010000
0040000
0004000
000009
000090

G:=sub<GL(6,GF(41))| [40,23,0,0,0,0,32,1,0,0,0,0,0,0,0,40,0,0,0,0,1,34,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[9,39,0,0,0,0,0,32,0,0,0,0,0,0,32,11,0,0,0,0,30,9,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,30,0,0,0,0,26,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,9,0,0,0,0,9,0] >;

Dic106Q8 in GAP, Magma, Sage, TeX

{\rm Dic}_{10}\rtimes_6Q_8
% in TeX

G:=Group("Dic10:6Q8");
// GroupNames label

G:=SmallGroup(320,721);
// by ID

G=gap.SmallGroup(320,721);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,120,254,219,100,1123,297,136,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^20=c^4=1,b^2=a^10,d^2=c^2,b*a*b^-1=a^-1,c*a*c^-1=a^11,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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