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G = Dic5.M4(2)  order 320 = 26·5

3rd non-split extension by Dic5 of M4(2) acting via M4(2)/C4=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic5.3M4(2), C5⋊C81Q8, C4⋊C4.8F5, C2.4(Q8×F5), C51(C84Q8), C10.2(C4×Q8), C20⋊C8.11C2, C10.11(C8○D4), Dic5.29(C2×Q8), (C2×Dic10).8C4, C2.12(D4.F5), Dic5⋊C8.5C2, C10.D4.10C4, C10.15(C2×M4(2)), Dic5.68(C4○D4), Dic53Q8.16C2, C22.79(C22×F5), C2.14(D5⋊M4(2)), C10.C42.3C2, (C2×Dic5).334C23, (C4×Dic5).250C22, (C4×C5⋊C8).9C2, (C5×C4⋊C4).11C4, (C2×C4).27(C2×F5), (C2×C20).85(C2×C4), (C2×C5⋊C8).31C22, (C2×C10).45(C22×C4), (C2×Dic5).60(C2×C4), SmallGroup(320,1045)

Series: Derived Chief Lower central Upper central

C1C2×C10 — Dic5.M4(2)
C1C5C10Dic5C2×Dic5C2×C5⋊C8C4×C5⋊C8 — Dic5.M4(2)
C5C2×C10 — Dic5.M4(2)
C1C22C4⋊C4

Generators and relations for Dic5.M4(2)
 G = < a,b,c,d | a10=c8=1, b2=d2=a5, bab-1=a-1, cac-1=a3, ad=da, cbc-1=dbd-1=a5b, dcd-1=a5c5 >

Subgroups: 282 in 94 conjugacy classes, 46 normal (42 characteristic)
C1, C2 [×3], C4 [×9], C22, C5, C8 [×5], C2×C4 [×3], C2×C4 [×4], Q8 [×2], C10 [×3], C42 [×3], C4⋊C4, C4⋊C4 [×2], C2×C8 [×4], C2×Q8, Dic5 [×2], Dic5 [×2], Dic5 [×2], C20 [×3], C2×C10, C4×C8, C8⋊C4 [×2], C4⋊C8 [×3], C4×Q8, C5⋊C8 [×2], C5⋊C8 [×3], Dic10 [×2], C2×Dic5 [×4], C2×C20 [×3], C84Q8, C4×Dic5 [×3], C10.D4 [×2], C5×C4⋊C4, C2×C5⋊C8 [×4], C2×Dic10, C4×C5⋊C8, C20⋊C8, C10.C42 [×2], Dic5⋊C8 [×2], Dic53Q8, Dic5.M4(2)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], Q8 [×2], C23, M4(2) [×2], C22×C4, C2×Q8, C4○D4, F5, C4×Q8, C2×M4(2), C8○D4, C2×F5 [×3], C84Q8, C22×F5, D5⋊M4(2), D4.F5, Q8×F5, Dic5.M4(2)

Smallest permutation representation of Dic5.M4(2)
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)
(1 115 6 120)(2 114 7 119)(3 113 8 118)(4 112 9 117)(5 111 10 116)(11 221 16 226)(12 230 17 225)(13 229 18 224)(14 228 19 223)(15 227 20 222)(21 129 26 124)(22 128 27 123)(23 127 28 122)(24 126 29 121)(25 125 30 130)(31 109 36 104)(32 108 37 103)(33 107 38 102)(34 106 39 101)(35 105 40 110)(41 94 46 99)(42 93 47 98)(43 92 48 97)(44 91 49 96)(45 100 50 95)(51 154 56 159)(52 153 57 158)(53 152 58 157)(54 151 59 156)(55 160 60 155)(61 168 66 163)(62 167 67 162)(63 166 68 161)(64 165 69 170)(65 164 70 169)(71 149 76 144)(72 148 77 143)(73 147 78 142)(74 146 79 141)(75 145 80 150)(81 134 86 139)(82 133 87 138)(83 132 88 137)(84 131 89 136)(85 140 90 135)(171 274 176 279)(172 273 177 278)(173 272 178 277)(174 271 179 276)(175 280 180 275)(181 289 186 284)(182 288 187 283)(183 287 188 282)(184 286 189 281)(185 285 190 290)(191 268 196 263)(192 267 197 262)(193 266 198 261)(194 265 199 270)(195 264 200 269)(201 255 206 260)(202 254 207 259)(203 253 208 258)(204 252 209 257)(205 251 210 256)(211 314 216 319)(212 313 217 318)(213 312 218 317)(214 311 219 316)(215 320 220 315)(231 308 236 303)(232 307 237 302)(233 306 238 301)(234 305 239 310)(235 304 240 309)(241 295 246 300)(242 294 247 299)(243 293 248 298)(244 292 249 297)(245 291 250 296)
(1 240 40 224 25 241 50 220)(2 237 39 227 26 248 49 213)(3 234 38 230 27 245 48 216)(4 231 37 223 28 242 47 219)(5 238 36 226 29 249 46 212)(6 235 35 229 30 246 45 215)(7 232 34 222 21 243 44 218)(8 239 33 225 22 250 43 211)(9 236 32 228 23 247 42 214)(10 233 31 221 24 244 41 217)(11 126 297 94 313 116 301 109)(12 123 296 97 314 113 310 102)(13 130 295 100 315 120 309 105)(14 127 294 93 316 117 308 108)(15 124 293 96 317 114 307 101)(16 121 292 99 318 111 306 104)(17 128 291 92 319 118 305 107)(18 125 300 95 320 115 304 110)(19 122 299 98 311 112 303 103)(20 129 298 91 312 119 302 106)(51 192 79 182 66 203 89 178)(52 199 78 185 67 210 88 171)(53 196 77 188 68 207 87 174)(54 193 76 181 69 204 86 177)(55 200 75 184 70 201 85 180)(56 197 74 187 61 208 84 173)(57 194 73 190 62 205 83 176)(58 191 72 183 63 202 82 179)(59 198 71 186 64 209 81 172)(60 195 80 189 65 206 90 175)(131 277 159 267 146 288 168 253)(132 274 158 270 147 285 167 256)(133 271 157 263 148 282 166 259)(134 278 156 266 149 289 165 252)(135 275 155 269 150 286 164 255)(136 272 154 262 141 283 163 258)(137 279 153 265 142 290 162 251)(138 276 152 268 143 287 161 254)(139 273 151 261 144 284 170 257)(140 280 160 264 145 281 169 260)
(1 55 6 60)(2 56 7 51)(3 57 8 52)(4 58 9 53)(5 59 10 54)(11 273 16 278)(12 274 17 279)(13 275 18 280)(14 276 19 271)(15 277 20 272)(21 66 26 61)(22 67 27 62)(23 68 28 63)(24 69 29 64)(25 70 30 65)(31 76 36 71)(32 77 37 72)(33 78 38 73)(34 79 39 74)(35 80 40 75)(41 86 46 81)(42 87 47 82)(43 88 48 83)(44 89 49 84)(45 90 50 85)(91 131 96 136)(92 132 97 137)(93 133 98 138)(94 134 99 139)(95 135 100 140)(101 141 106 146)(102 142 107 147)(103 143 108 148)(104 144 109 149)(105 145 110 150)(111 151 116 156)(112 152 117 157)(113 153 118 158)(114 154 119 159)(115 155 120 160)(121 170 126 165)(122 161 127 166)(123 162 128 167)(124 163 129 168)(125 164 130 169)(171 225 176 230)(172 226 177 221)(173 227 178 222)(174 228 179 223)(175 229 180 224)(181 217 186 212)(182 218 187 213)(183 219 188 214)(184 220 189 215)(185 211 190 216)(191 242 196 247)(192 243 197 248)(193 244 198 249)(194 245 199 250)(195 246 200 241)(201 240 206 235)(202 231 207 236)(203 232 208 237)(204 233 209 238)(205 234 210 239)(251 310 256 305)(252 301 257 306)(253 302 258 307)(254 303 259 308)(255 304 260 309)(261 292 266 297)(262 293 267 298)(263 294 268 299)(264 295 269 300)(265 296 270 291)(281 315 286 320)(282 316 287 311)(283 317 288 312)(284 318 289 313)(285 319 290 314)

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

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

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

38 conjugacy classes

class 1 2A2B2C4A4B4C4D4E4F4G4H4I4J4K4L 5 8A···8H8I8J8K8L10A10B10C20A···20F
order122244444444444458···8888810101020···20
size11112244555510102020410···10202020204448···8

38 irreducible representations

dim111111111222244488
type++++++-++--
imageC1C2C2C2C2C2C4C4C4Q8M4(2)C4○D4C8○D4F5C2×F5D5⋊M4(2)D4.F5Q8×F5
kernelDic5.M4(2)C4×C5⋊C8C20⋊C8C10.C42Dic5⋊C8Dic53Q8C10.D4C5×C4⋊C4C2×Dic10C5⋊C8Dic5Dic5C10C4⋊C4C2×C4C2C2C2
# reps111221422242413411

Matrix representation of Dic5.M4(2) in GL8(𝔽41)

400000000
040000000
00100000
00010000
000040100
000040010
000040001
000040000
,
130000000
3040000000
004000000
000400000
0000193038
00002233819
00002201938
00001922380
,
2432000000
3217000000
009210000
0020320000
00003327327
000036133619
00002852222
0000148814
,
01000000
400000000
00010000
00100000
00001000
00000100
00000010
00000001

G:=sub<GL(8,GF(41))| [40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,40,40,40,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0],[1,30,0,0,0,0,0,0,30,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,19,22,22,19,0,0,0,0,3,3,0,22,0,0,0,0,0,38,19,38,0,0,0,0,38,19,38,0],[24,32,0,0,0,0,0,0,32,17,0,0,0,0,0,0,0,0,9,20,0,0,0,0,0,0,21,32,0,0,0,0,0,0,0,0,33,36,28,14,0,0,0,0,27,13,5,8,0,0,0,0,3,36,22,8,0,0,0,0,27,19,22,14],[0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1] >;

Dic5.M4(2) in GAP, Magma, Sage, TeX

{\rm Dic}_5.M_4(2)
% in TeX

G:=Group("Dic5.M4(2)");
// GroupNames label

G:=SmallGroup(320,1045);
// by ID

G=gap.SmallGroup(320,1045);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,344,1094,219,100,136,6278,1595]);
// Polycyclic

G:=Group<a,b,c,d|a^10=c^8=1,b^2=d^2=a^5,b*a*b^-1=a^-1,c*a*c^-1=a^3,a*d=d*a,c*b*c^-1=d*b*d^-1=a^5*b,d*c*d^-1=a^5*c^5>;
// generators/relations

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