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G = Dic105Q8order 320 = 26·5

3rd semidirect product of Dic10 and Q8 acting via Q8/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic105Q8, C20.10Q16, C42.84D10, C4⋊Q8.10D5, C4.13(Q8×D5), C4⋊C4.86D10, C55(C4.Q16), C20.40(C2×Q8), (C2×C20).160D4, C10.43(C2×Q16), C203C8.22C2, C4.8(C5⋊Q16), C20.86(C4○D4), (C2×C20).409C23, (C4×C20).138C22, C4.36(Q82D5), (C4×Dic10).18C2, C10.D8.17C2, C10.Q16.14C2, C10.77(C22⋊Q8), C10.100(C8⋊C22), C2.14(D103Q8), C4⋊Dic5.350C22, C2.21(D4.D10), (C2×Dic10).284C22, (C5×C4⋊Q8).10C2, C2.14(C2×C5⋊Q16), (C2×C10).540(C2×D4), (C2×C4).191(C5⋊D4), (C5×C4⋊C4).133C22, (C2×C4).506(C22×D5), C22.212(C2×C5⋊D4), (C2×C52C8).141C22, SmallGroup(320,718)

Series: Derived Chief Lower central Upper central

C1C2×C20 — Dic105Q8
C1C5C10C20C2×C20C2×Dic10C4×Dic10 — Dic105Q8
C5C10C2×C20 — Dic105Q8
C1C22C42C4⋊Q8

Generators and relations for Dic105Q8
 G = < a,b,c,d | a20=c4=1, b2=a10, d2=c2, bab-1=a-1, cac-1=a11, ad=da, cbc-1=a5b, dbd-1=a10b, 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, C2.D8 [×2], C4×Q8, C4⋊Q8, C52C8 [×2], Dic10 [×2], Dic10, C2×Dic5 [×2], C2×C20 [×3], C2×C20 [×2], C5×Q8 [×2], C4.Q16, C2×C52C8 [×2], C4×Dic5, C10.D4, C4⋊Dic5, C4×C20, C5×C4⋊C4 [×2], C5×C4⋊C4, C2×Dic10, Q8×C10, C203C8, C10.D8 [×2], C10.Q16 [×2], C4×Dic10, C5×C4⋊Q8, Dic105Q8
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], Q8 [×2], C23, D5, Q16 [×2], C2×D4, C2×Q8, C4○D4, D10 [×3], C22⋊Q8, C2×Q16, C8⋊C22, C5⋊D4 [×2], C22×D5, C4.Q16, C5⋊Q16 [×2], Q8×D5, Q82D5, C2×C5⋊D4, D4.D10, C2×C5⋊Q16, D103Q8, Dic105Q8

Smallest permutation representation of Dic105Q8
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)
(1 236 11 226)(2 235 12 225)(3 234 13 224)(4 233 14 223)(5 232 15 222)(6 231 16 221)(7 230 17 240)(8 229 18 239)(9 228 19 238)(10 227 20 237)(21 125 31 135)(22 124 32 134)(23 123 33 133)(24 122 34 132)(25 121 35 131)(26 140 36 130)(27 139 37 129)(28 138 38 128)(29 137 39 127)(30 136 40 126)(41 206 51 216)(42 205 52 215)(43 204 53 214)(44 203 54 213)(45 202 55 212)(46 201 56 211)(47 220 57 210)(48 219 58 209)(49 218 59 208)(50 217 60 207)(61 160 71 150)(62 159 72 149)(63 158 73 148)(64 157 74 147)(65 156 75 146)(66 155 76 145)(67 154 77 144)(68 153 78 143)(69 152 79 142)(70 151 80 141)(81 175 91 165)(82 174 92 164)(83 173 93 163)(84 172 94 162)(85 171 95 161)(86 170 96 180)(87 169 97 179)(88 168 98 178)(89 167 99 177)(90 166 100 176)(101 182 111 192)(102 181 112 191)(103 200 113 190)(104 199 114 189)(105 198 115 188)(106 197 116 187)(107 196 117 186)(108 195 118 185)(109 194 119 184)(110 193 120 183)(241 310 251 320)(242 309 252 319)(243 308 253 318)(244 307 254 317)(245 306 255 316)(246 305 256 315)(247 304 257 314)(248 303 258 313)(249 302 259 312)(250 301 260 311)(261 287 271 297)(262 286 272 296)(263 285 273 295)(264 284 274 294)(265 283 275 293)(266 282 276 292)(267 281 277 291)(268 300 278 290)(269 299 279 289)(270 298 280 288)
(1 26 211 278)(2 37 212 269)(3 28 213 280)(4 39 214 271)(5 30 215 262)(6 21 216 273)(7 32 217 264)(8 23 218 275)(9 34 219 266)(10 25 220 277)(11 36 201 268)(12 27 202 279)(13 38 203 270)(14 29 204 261)(15 40 205 272)(16 31 206 263)(17 22 207 274)(18 33 208 265)(19 24 209 276)(20 35 210 267)(41 290 231 140)(42 281 232 131)(43 292 233 122)(44 283 234 133)(45 294 235 124)(46 285 236 135)(47 296 237 126)(48 287 238 137)(49 298 239 128)(50 289 240 139)(51 300 221 130)(52 291 222 121)(53 282 223 132)(54 293 224 123)(55 284 225 134)(56 295 226 125)(57 286 227 136)(58 297 228 127)(59 288 229 138)(60 299 230 129)(61 317 99 191)(62 308 100 182)(63 319 81 193)(64 310 82 184)(65 301 83 195)(66 312 84 186)(67 303 85 197)(68 314 86 188)(69 305 87 199)(70 316 88 190)(71 307 89 181)(72 318 90 192)(73 309 91 183)(74 320 92 194)(75 311 93 185)(76 302 94 196)(77 313 95 187)(78 304 96 198)(79 315 97 189)(80 306 98 200)(101 144 243 161)(102 155 244 172)(103 146 245 163)(104 157 246 174)(105 148 247 165)(106 159 248 176)(107 150 249 167)(108 141 250 178)(109 152 251 169)(110 143 252 180)(111 154 253 171)(112 145 254 162)(113 156 255 173)(114 147 256 164)(115 158 257 175)(116 149 258 166)(117 160 259 177)(118 151 260 168)(119 142 241 179)(120 153 242 170)
(1 83 211 65)(2 84 212 66)(3 85 213 67)(4 86 214 68)(5 87 215 69)(6 88 216 70)(7 89 217 71)(8 90 218 72)(9 91 219 73)(10 92 220 74)(11 93 201 75)(12 94 202 76)(13 95 203 77)(14 96 204 78)(15 97 205 79)(16 98 206 80)(17 99 207 61)(18 100 208 62)(19 81 209 63)(20 82 210 64)(21 316 273 190)(22 317 274 191)(23 318 275 192)(24 319 276 193)(25 320 277 194)(26 301 278 195)(27 302 279 196)(28 303 280 197)(29 304 261 198)(30 305 262 199)(31 306 263 200)(32 307 264 181)(33 308 265 182)(34 309 266 183)(35 310 267 184)(36 311 268 185)(37 312 269 186)(38 313 270 187)(39 314 271 188)(40 315 272 189)(41 141 231 178)(42 142 232 179)(43 143 233 180)(44 144 234 161)(45 145 235 162)(46 146 236 163)(47 147 237 164)(48 148 238 165)(49 149 239 166)(50 150 240 167)(51 151 221 168)(52 152 222 169)(53 153 223 170)(54 154 224 171)(55 155 225 172)(56 156 226 173)(57 157 227 174)(58 158 228 175)(59 159 229 176)(60 160 230 177)(101 133 243 283)(102 134 244 284)(103 135 245 285)(104 136 246 286)(105 137 247 287)(106 138 248 288)(107 139 249 289)(108 140 250 290)(109 121 251 291)(110 122 252 292)(111 123 253 293)(112 124 254 294)(113 125 255 295)(114 126 256 296)(115 127 257 297)(116 128 258 298)(117 129 259 299)(118 130 260 300)(119 131 241 281)(120 132 242 282)

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

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

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

47 conjugacy classes

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

47 irreducible representations

dim1111112222222244444
type++++++-++-+++--+
imageC1C2C2C2C2C2Q8D4D5Q16C4○D4D10D10C5⋊D4C8⋊C22C5⋊Q16Q8×D5Q82D5D4.D10
kernelDic105Q8C203C8C10.D8C10.Q16C4×Dic10C5×C4⋊Q8Dic10C2×C20C4⋊Q8C20C20C42C4⋊C4C2×C4C10C4C4C4C2
# reps1122112224224814224

Matrix representation of Dic105Q8 in GL6(𝔽41)

010000
4060000
0013900
0014000
000010
000001
,
4060000
010000
0017100
00382400
000010
000001
,
4000000
0400000
0010200
00113100
0000259
00001716
,
4000000
0400000
0013900
0014000
0000153
00003426

G:=sub<GL(6,GF(41))| [0,40,0,0,0,0,1,6,0,0,0,0,0,0,1,1,0,0,0,0,39,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,6,1,0,0,0,0,0,0,17,38,0,0,0,0,1,24,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,10,11,0,0,0,0,2,31,0,0,0,0,0,0,25,17,0,0,0,0,9,16],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,1,0,0,0,0,39,40,0,0,0,0,0,0,15,34,0,0,0,0,3,26] >;

Dic105Q8 in GAP, Magma, Sage, TeX

{\rm Dic}_{10}\rtimes_5Q_8
% in TeX

G:=Group("Dic10:5Q8");
// GroupNames label

G:=SmallGroup(320,718);
// by ID

G=gap.SmallGroup(320,718);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,120,254,219,268,1123,297,136,12550]);
// 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,d*b*d^-1=a^10*b,d*c*d^-1=c^-1>;
// generators/relations

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