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G = Dic102Q8order 320 = 26·5

2nd semidirect product of Dic10 and Q8 acting via Q8/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic102Q8, Dic5.10Q16, C4.5(Q8×D5), C20⋊Q8.9C2, C53(C4.Q16), C2.D8.5D5, (C2×C8).25D10, C2.13(D5×Q16), C20.18(C2×Q8), C4⋊C4.168D10, C10.22(C2×Q16), C4.78(C4○D20), C10.Q16.8C2, C10.D8.8C2, C22.224(D4×D5), C20.170(C4○D4), C2.20(D8⋊D5), C10.38(C8⋊C22), (C2×C40).239C22, (C2×C20).291C23, Dic53Q8.8C2, (C2×Dic5).222D4, C10.39(C22⋊Q8), C20.8Q8.10C2, C2.16(D10⋊Q8), C20.44D4.10C2, C4⋊Dic5.117C22, (C4×Dic5).40C22, (C2×Dic10).91C22, (C5×C2.D8).11C2, (C2×C10).296(C2×D4), (C5×C4⋊C4).84C22, (C2×C52C8).65C22, (C2×C4).394(C22×D5), SmallGroup(320,502)

Series: Derived Chief Lower central Upper central

C1C2×C20 — Dic102Q8
C1C5C10C20C2×C20C4×Dic5Dic53Q8 — Dic102Q8
C5C10C2×C20 — Dic102Q8
C1C22C2×C4C2.D8

Generators and relations for Dic102Q8
 G = < a,b,c,d | a20=c4=1, b2=a10, d2=c2, bab-1=a-1, cac-1=a11, dad-1=a9, cbc-1=a5b, dbd-1=a10b, dcd-1=c-1 >

Subgroups: 358 in 96 conjugacy classes, 41 normal (37 characteristic)
C1, C2 [×3], C4 [×2], C4 [×8], C22, C5, C8 [×2], C2×C4, C2×C4 [×6], Q8 [×5], C10 [×3], C42 [×2], C4⋊C4 [×2], C4⋊C4 [×3], C2×C8, C2×C8, C2×Q8 [×2], Dic5 [×2], Dic5 [×4], C20 [×2], C20 [×2], C2×C10, Q8⋊C4 [×2], C4⋊C8, C2.D8, C2.D8, C4×Q8, C4⋊Q8, C52C8, C40, Dic10 [×2], Dic10 [×3], C2×Dic5 [×2], C2×Dic5 [×2], C2×C20, C2×C20 [×2], C4.Q16, C2×C52C8, C4×Dic5, C4×Dic5, C10.D4 [×2], C4⋊Dic5, C5×C4⋊C4 [×2], C2×C40, C2×Dic10, C2×Dic10, C10.D8, C10.Q16, C20.8Q8, C20.44D4, C5×C2.D8, Dic53Q8, C20⋊Q8, Dic102Q8
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, C22×D5, C4.Q16, C4○D20, D4×D5, Q8×D5, D10⋊Q8, D8⋊D5, D5×Q16, Dic102Q8

Smallest permutation representation of Dic102Q8
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 94 11 84)(2 93 12 83)(3 92 13 82)(4 91 14 81)(5 90 15 100)(6 89 16 99)(7 88 17 98)(8 87 18 97)(9 86 19 96)(10 85 20 95)(21 263 31 273)(22 262 32 272)(23 261 33 271)(24 280 34 270)(25 279 35 269)(26 278 36 268)(27 277 37 267)(28 276 38 266)(29 275 39 265)(30 274 40 264)(41 157 51 147)(42 156 52 146)(43 155 53 145)(44 154 54 144)(45 153 55 143)(46 152 56 142)(47 151 57 141)(48 150 58 160)(49 149 59 159)(50 148 60 158)(61 238 71 228)(62 237 72 227)(63 236 73 226)(64 235 74 225)(65 234 75 224)(66 233 76 223)(67 232 77 222)(68 231 78 221)(69 230 79 240)(70 229 80 239)(101 175 111 165)(102 174 112 164)(103 173 113 163)(104 172 114 162)(105 171 115 161)(106 170 116 180)(107 169 117 179)(108 168 118 178)(109 167 119 177)(110 166 120 176)(121 284 131 294)(122 283 132 293)(123 282 133 292)(124 281 134 291)(125 300 135 290)(126 299 136 289)(127 298 137 288)(128 297 138 287)(129 296 139 286)(130 295 140 285)(181 260 191 250)(182 259 192 249)(183 258 193 248)(184 257 194 247)(185 256 195 246)(186 255 196 245)(187 254 197 244)(188 253 198 243)(189 252 199 242)(190 251 200 241)(201 310 211 320)(202 309 212 319)(203 308 213 318)(204 307 214 317)(205 306 215 316)(206 305 216 315)(207 304 217 314)(208 303 218 313)(209 302 219 312)(210 301 220 311)
(1 293 312 173)(2 284 313 164)(3 295 314 175)(4 286 315 166)(5 297 316 177)(6 288 317 168)(7 299 318 179)(8 290 319 170)(9 281 320 161)(10 292 301 172)(11 283 302 163)(12 294 303 174)(13 285 304 165)(14 296 305 176)(15 287 306 167)(16 298 307 178)(17 289 308 169)(18 300 309 180)(19 291 310 171)(20 282 311 162)(21 156 66 250)(22 147 67 241)(23 158 68 252)(24 149 69 243)(25 160 70 254)(26 151 71 245)(27 142 72 256)(28 153 73 247)(29 144 74 258)(30 155 75 249)(31 146 76 260)(32 157 77 251)(33 148 78 242)(34 159 79 253)(35 150 80 244)(36 141 61 255)(37 152 62 246)(38 143 63 257)(39 154 64 248)(40 145 65 259)(41 227 190 277)(42 238 191 268)(43 229 192 279)(44 240 193 270)(45 231 194 261)(46 222 195 272)(47 233 196 263)(48 224 197 274)(49 235 198 265)(50 226 199 276)(51 237 200 267)(52 228 181 278)(53 239 182 269)(54 230 183 280)(55 221 184 271)(56 232 185 262)(57 223 186 273)(58 234 187 264)(59 225 188 275)(60 236 189 266)(81 134 216 105)(82 125 217 116)(83 136 218 107)(84 127 219 118)(85 138 220 109)(86 129 201 120)(87 140 202 111)(88 131 203 102)(89 122 204 113)(90 133 205 104)(91 124 206 115)(92 135 207 106)(93 126 208 117)(94 137 209 108)(95 128 210 119)(96 139 211 110)(97 130 212 101)(98 121 213 112)(99 132 214 103)(100 123 215 114)
(1 268 312 238)(2 277 313 227)(3 266 314 236)(4 275 315 225)(5 264 316 234)(6 273 317 223)(7 262 318 232)(8 271 319 221)(9 280 320 230)(10 269 301 239)(11 278 302 228)(12 267 303 237)(13 276 304 226)(14 265 305 235)(15 274 306 224)(16 263 307 233)(17 272 308 222)(18 261 309 231)(19 270 310 240)(20 279 311 229)(21 214 66 99)(22 203 67 88)(23 212 68 97)(24 201 69 86)(25 210 70 95)(26 219 71 84)(27 208 72 93)(28 217 73 82)(29 206 74 91)(30 215 75 100)(31 204 76 89)(32 213 77 98)(33 202 78 87)(34 211 79 96)(35 220 80 85)(36 209 61 94)(37 218 62 83)(38 207 63 92)(39 216 64 81)(40 205 65 90)(41 284 190 164)(42 293 191 173)(43 282 192 162)(44 291 193 171)(45 300 194 180)(46 289 195 169)(47 298 196 178)(48 287 197 167)(49 296 198 176)(50 285 199 165)(51 294 200 174)(52 283 181 163)(53 292 182 172)(54 281 183 161)(55 290 184 170)(56 299 185 179)(57 288 186 168)(58 297 187 177)(59 286 188 166)(60 295 189 175)(101 158 130 252)(102 147 131 241)(103 156 132 250)(104 145 133 259)(105 154 134 248)(106 143 135 257)(107 152 136 246)(108 141 137 255)(109 150 138 244)(110 159 139 253)(111 148 140 242)(112 157 121 251)(113 146 122 260)(114 155 123 249)(115 144 124 258)(116 153 125 247)(117 142 126 256)(118 151 127 245)(119 160 128 254)(120 149 129 243)

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

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

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

47 conjugacy classes

class 1 2A2B2C4A4B4C4D4E4F4G4H4I4J4K5A5B8A8B8C8D10A···10F20A20B20C20D20E···20L40A···40H
order12224444444444455888810···102020202020···2040···40
size111122448101020202040224420202···244448···84···4

47 irreducible representations

dim111111112222222244444
type++++++++-++-+++-+-
imageC1C2C2C2C2C2C2C2Q8D4D5Q16C4○D4D10D10C4○D20C8⋊C22Q8×D5D4×D5D8⋊D5D5×Q16
kernelDic102Q8C10.D8C10.Q16C20.8Q8C20.44D4C5×C2.D8Dic53Q8C20⋊Q8Dic10C2×Dic5C2.D8Dic5C20C4⋊C4C2×C8C4C10C4C22C2C2
# reps111111112224242812244

Matrix representation of Dic102Q8 in GL6(𝔽41)

100000
010000
006100
0040000
000001
0000400
,
100000
010000
0063500
00403500
0000320
00002038
,
2050000
10210000
001000
000100
0000130
00003040
,
3150000
29100000
0035600
001600
000001
0000400

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,6,40,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,6,40,0,0,0,0,35,35,0,0,0,0,0,0,3,20,0,0,0,0,20,38],[20,10,0,0,0,0,5,21,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,30,0,0,0,0,30,40],[31,29,0,0,0,0,5,10,0,0,0,0,0,0,35,1,0,0,0,0,6,6,0,0,0,0,0,0,0,40,0,0,0,0,1,0] >;

Dic102Q8 in GAP, Magma, Sage, TeX

{\rm Dic}_{10}\rtimes_2Q_8
% in TeX

G:=Group("Dic10:2Q8");
// GroupNames label

G:=SmallGroup(320,502);
// by ID

G=gap.SmallGroup(320,502);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,344,422,135,268,570,297,136,12550]);
// 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,d*a*d^-1=a^9,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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