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## G = Dic10⋊7Q8order 320 = 26·5

### 5th semidirect product of Dic10 and Q8 acting via Q8/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — Dic10⋊7Q8
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C4×Dic5 — Dic5⋊3Q8 — Dic10⋊7Q8
 Lower central C5 — C2×C10 — Dic10⋊7Q8
 Upper central C1 — C22 — C42.C2

Generators and relations for Dic107Q8
G = < a,b,c,d | a20=c4=1, b2=a10, d2=c2, bab-1=a-1, ac=ca, dad-1=a9, cbc-1=dbd-1=a10b, dcd-1=c-1 >

Subgroups: 566 in 200 conjugacy classes, 105 normal (43 characteristic)
C1, C2 [×3], C4 [×2], C4 [×17], C22, C5, C2×C4 [×3], C2×C4 [×4], C2×C4 [×8], Q8 [×10], C10 [×3], C42, C42 [×8], C4⋊C4 [×2], C4⋊C4 [×4], C4⋊C4 [×16], C2×Q8 [×4], Dic5 [×6], Dic5 [×5], C20 [×2], C20 [×6], C2×C10, C4×Q8 [×6], C42.C2, C42.C2 [×5], C4⋊Q8 [×3], Dic10 [×4], Dic10 [×6], C2×Dic5 [×4], C2×Dic5 [×4], C2×C20 [×3], C2×C20 [×4], Q83Q8, C4×Dic5 [×2], C4×Dic5 [×6], C10.D4 [×2], C10.D4 [×10], C4⋊Dic5 [×2], C4⋊Dic5 [×2], C4×C20, C5×C4⋊C4 [×2], C5×C4⋊C4 [×4], C2×Dic10 [×2], C2×Dic10 [×2], C4×Dic10 [×2], Dic53Q8 [×2], Dic53Q8 [×2], C20⋊Q8, C20⋊Q8 [×2], Dic5.Q8 [×4], C4.Dic10, C5×C42.C2, Dic107Q8
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, Q8×D5 [×2], C23×D5, C2×Q8×D5, D5×C4○D4, D4.10D10, Dic107Q8

Smallest permutation representation of Dic107Q8
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)
(1 182 11 192)(2 181 12 191)(3 200 13 190)(4 199 14 189)(5 198 15 188)(6 197 16 187)(7 196 17 186)(8 195 18 185)(9 194 19 184)(10 193 20 183)(21 49 31 59)(22 48 32 58)(23 47 33 57)(24 46 34 56)(25 45 35 55)(26 44 36 54)(27 43 37 53)(28 42 38 52)(29 41 39 51)(30 60 40 50)(61 150 71 160)(62 149 72 159)(63 148 73 158)(64 147 74 157)(65 146 75 156)(66 145 76 155)(67 144 77 154)(68 143 78 153)(69 142 79 152)(70 141 80 151)(81 314 91 304)(82 313 92 303)(83 312 93 302)(84 311 94 301)(85 310 95 320)(86 309 96 319)(87 308 97 318)(88 307 98 317)(89 306 99 316)(90 305 100 315)(101 263 111 273)(102 262 112 272)(103 261 113 271)(104 280 114 270)(105 279 115 269)(106 278 116 268)(107 277 117 267)(108 276 118 266)(109 275 119 265)(110 274 120 264)(121 214 131 204)(122 213 132 203)(123 212 133 202)(124 211 134 201)(125 210 135 220)(126 209 136 219)(127 208 137 218)(128 207 138 217)(129 206 139 216)(130 205 140 215)(161 238 171 228)(162 237 172 227)(163 236 173 226)(164 235 174 225)(165 234 175 224)(166 233 176 223)(167 232 177 222)(168 231 178 221)(169 230 179 240)(170 229 180 239)(241 284 251 294)(242 283 252 293)(243 282 253 292)(244 281 254 291)(245 300 255 290)(246 299 256 289)(247 298 257 288)(248 297 258 287)(249 296 259 286)(250 295 260 285)
(1 134 256 113)(2 135 257 114)(3 136 258 115)(4 137 259 116)(5 138 260 117)(6 139 241 118)(7 140 242 119)(8 121 243 120)(9 122 244 101)(10 123 245 102)(11 124 246 103)(12 125 247 104)(13 126 248 105)(14 127 249 106)(15 128 250 107)(16 129 251 108)(17 130 252 109)(18 131 253 110)(19 132 254 111)(20 133 255 112)(21 86 229 70)(22 87 230 71)(23 88 231 72)(24 89 232 73)(25 90 233 74)(26 91 234 75)(27 92 235 76)(28 93 236 77)(29 94 237 78)(30 95 238 79)(31 96 239 80)(32 97 240 61)(33 98 221 62)(34 99 222 63)(35 100 223 64)(36 81 224 65)(37 82 225 66)(38 83 226 67)(39 84 227 68)(40 85 228 69)(41 311 172 143)(42 312 173 144)(43 313 174 145)(44 314 175 146)(45 315 176 147)(46 316 177 148)(47 317 178 149)(48 318 179 150)(49 319 180 151)(50 320 161 152)(51 301 162 153)(52 302 163 154)(53 303 164 155)(54 304 165 156)(55 305 166 157)(56 306 167 158)(57 307 168 159)(58 308 169 160)(59 309 170 141)(60 310 171 142)(181 210 288 280)(182 211 289 261)(183 212 290 262)(184 213 291 263)(185 214 292 264)(186 215 293 265)(187 216 294 266)(188 217 295 267)(189 218 296 268)(190 219 297 269)(191 220 298 270)(192 201 299 271)(193 202 300 272)(194 203 281 273)(195 204 282 274)(196 205 283 275)(197 206 284 276)(198 207 285 277)(199 208 286 278)(200 209 287 279)
(1 235 256 27)(2 224 257 36)(3 233 258 25)(4 222 259 34)(5 231 260 23)(6 240 241 32)(7 229 242 21)(8 238 243 30)(9 227 244 39)(10 236 245 28)(11 225 246 37)(12 234 247 26)(13 223 248 35)(14 232 249 24)(15 221 250 33)(16 230 251 22)(17 239 252 31)(18 228 253 40)(19 237 254 29)(20 226 255 38)(41 194 172 281)(42 183 173 290)(43 192 174 299)(44 181 175 288)(45 190 176 297)(46 199 177 286)(47 188 178 295)(48 197 179 284)(49 186 180 293)(50 195 161 282)(51 184 162 291)(52 193 163 300)(53 182 164 289)(54 191 165 298)(55 200 166 287)(56 189 167 296)(57 198 168 285)(58 187 169 294)(59 196 170 283)(60 185 171 292)(61 139 97 118)(62 128 98 107)(63 137 99 116)(64 126 100 105)(65 135 81 114)(66 124 82 103)(67 133 83 112)(68 122 84 101)(69 131 85 110)(70 140 86 119)(71 129 87 108)(72 138 88 117)(73 127 89 106)(74 136 90 115)(75 125 91 104)(76 134 92 113)(77 123 93 102)(78 132 94 111)(79 121 95 120)(80 130 96 109)(141 205 309 275)(142 214 310 264)(143 203 311 273)(144 212 312 262)(145 201 313 271)(146 210 314 280)(147 219 315 269)(148 208 316 278)(149 217 317 267)(150 206 318 276)(151 215 319 265)(152 204 320 274)(153 213 301 263)(154 202 302 272)(155 211 303 261)(156 220 304 270)(157 209 305 279)(158 218 306 268)(159 207 307 277)(160 216 308 266)

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

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

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

53 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E ··· 4I 4J ··· 4Q 4R 4S 4T 4U 5A 5B 10A ··· 10F 20A ··· 20L 20M ··· 20T order 1 2 2 2 4 4 4 4 4 ··· 4 4 ··· 4 4 4 4 4 5 5 10 ··· 10 20 ··· 20 20 ··· 20 size 1 1 1 1 2 2 2 2 4 ··· 4 10 ··· 10 20 20 20 20 2 2 2 ··· 2 4 ··· 4 8 ··· 8

53 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 4 4 4 4 type + + + + + + + - + + + - - - image C1 C2 C2 C2 C2 C2 C2 Q8 D5 C4○D4 D10 D10 2- 1+4 Q8×D5 D5×C4○D4 D4.10D10 kernel Dic10⋊7Q8 C4×Dic10 Dic5⋊3Q8 C20⋊Q8 Dic5.Q8 C4.Dic10 C5×C42.C2 Dic10 C42.C2 Dic5 C42 C4⋊C4 C10 C4 C2 C2 # reps 1 2 4 3 4 1 1 4 2 4 2 12 1 4 4 4

Matrix representation of Dic107Q8 in GL6(𝔽41)

 0 32 0 0 0 0 32 0 0 0 0 0 0 0 31 0 0 0 0 0 31 4 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 39 28 0 0 0 0 13 2 0 0 0 0 0 0 38 37 0 0 0 0 2 3 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 40 0 0 0 0 40 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 40 0
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 38 37 0 0 0 0 2 3 0 0 0 0 0 0 2 35 0 0 0 0 35 39

G:=sub<GL(6,GF(41))| [0,32,0,0,0,0,32,0,0,0,0,0,0,0,31,31,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[39,13,0,0,0,0,28,2,0,0,0,0,0,0,38,2,0,0,0,0,37,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,40,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,1,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,38,2,0,0,0,0,37,3,0,0,0,0,0,0,2,35,0,0,0,0,35,39] >;

Dic107Q8 in GAP, Magma, Sage, TeX

{\rm Dic}_{10}\rtimes_7Q_8
% in TeX

G:=Group("Dic10:7Q8");
// GroupNames label

G:=SmallGroup(320,1357);
// by ID

G=gap.SmallGroup(320,1357);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,120,219,268,1571,297,192,12550]);
// 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,d*a*d^-1=a^9,c*b*c^-1=d*b*d^-1=a^10*b,d*c*d^-1=c^-1>;
// generators/relations

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