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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⋊C8⋊1Q8, C4⋊C4.8F5, C2.4(Q8×F5), C5⋊1(C8⋊4Q8), 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), Dic5⋊3Q8.16C2, 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

Derived series
C1 — C2×C10 — Dic5.M4(2)
Chief series
C1 — C5 — C10 — Dic5 — C2×Dic5 — C2×C5⋊C8 — C4×C5⋊C8 — Dic5.M4(2)
Lower central
C5 — C2×C10 — Dic5.M4(2)
Upper central
C1 — C22 — C4⋊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, C4, C22, C5, C8, C2×C4, C2×C4, Q8, C10, C42, C4⋊C4, C4⋊C4, C2×C8, C2×Q8, Dic5, Dic5, Dic5, C20, C2×C10, C4×C8, C8⋊C4, C4⋊C8, C4×Q8, C5⋊C8, C5⋊C8, Dic10, C2×Dic5, C2×C20, C8⋊4Q8, C4×Dic5, C10.D4, C5×C4⋊C4, C2×C5⋊C8, C2×Dic10, C4×C5⋊C8, C20⋊C8, C10.C42, Dic5⋊C8, Dic5⋊3Q8, Dic5.M4(2)
Quotients: C1, C2, C4, C22, C2×C4, Q8, C23, M4(2), C22×C4, C2×Q8, C4○D4, F5, C4×Q8, C2×M4(2), C8○D4, C2×F5, C8⋊4Q8, 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)

(1 119 6 114)(2 118 7 113)(3 117 8 112)(4 116 9 111)(5 115 10 120)(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)(31 103 36 108)(32 102 37 107)(33 101 38 106)(34 110 39 105)(35 109 40 104)(41 100 46 95)(42 99 47 94)(43 98 48 93)(44 97 49 92)(45 96 50 91)(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)(71 143 76 148)(72 142 77 147)(73 141 78 146)(74 150 79 145)(75 149 80 144)(81 140 86 135)(82 139 87 134)(83 138 88 133)(84 137 89 132)(85 136 90 131)(171 272 176 277)(172 271 177 276)(173 280 178 275)(174 279 179 274)(175 278 180 273)(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)(201 253 206 258)(202 252 207 257)(203 251 208 256)(204 260 209 255)(205 259 210 254)(211 312 216 317)(212 311 217 316)(213 320 218 315)(214 319 219 314)(215 318 220 313)(231 308 236 303)(232 307 237 302)(233 306 238 301)(234 305 239 310)(235 304 240 309)(241 293 246 298)(242 292 247 297)(243 291 248 296)(244 300 249 295)(245 299 250 294)

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

(1 51 6 56)(2 52 7 57)(3 53 8 58)(4 54 9 59)(5 55 10 60)(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)(171 227 176 222)(172 228 177 223)(173 229 178 224)(174 230 179 225)(175 221 180 226)(181 215 186 220)(182 216 187 211)(183 217 188 212)(184 218 189 213)(185 219 190 214)(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)(251 302 256 307)(252 303 257 308)(253 304 258 309)(254 305 259 310)(255 306 260 301)(261 300 266 295)(262 291 267 296)(263 292 268 297)(264 293 269 298)(265 294 270 299)(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), (1,119,6,114)(2,118,7,113)(3,117,8,112)(4,116,9,111)(5,115,10,120)(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)(31,103,36,108)(32,102,37,107)(33,101,38,106)(34,110,39,105)(35,109,40,104)(41,100,46,95)(42,99,47,94)(43,98,48,93)(44,97,49,92)(45,96,50,91)(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)(71,143,76,148)(72,142,77,147)(73,141,78,146)(74,150,79,145)(75,149,80,144)(81,140,86,135)(82,139,87,134)(83,138,88,133)(84,137,89,132)(85,136,90,131)(171,272,176,277)(172,271,177,276)(173,280,178,275)(174,279,179,274)(175,278,180,273)(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)(201,253,206,258)(202,252,207,257)(203,251,208,256)(204,260,209,255)(205,259,210,254)(211,312,216,317)(212,311,217,316)(213,320,218,315)(214,319,219,314)(215,318,220,313)(231,308,236,303)(232,307,237,302)(233,306,238,301)(234,305,239,310)(235,304,240,309)(241,293,246,298)(242,292,247,297)(243,291,248,296)(244,300,249,295)(245,299,250,294), (1,240,34,224,25,241,50,218)(2,237,33,227,26,248,49,211)(3,234,32,230,27,245,48,214)(4,231,31,223,28,242,47,217)(5,238,40,226,29,249,46,220)(6,235,39,229,30,246,45,213)(7,232,38,222,21,243,44,216)(8,239,37,225,22,250,43,219)(9,236,36,228,23,247,42,212)(10,233,35,221,24,244,41,215)(11,126,295,100,313,120,301,109)(12,123,294,93,314,117,310,102)(13,130,293,96,315,114,309,105)(14,127,292,99,316,111,308,108)(15,124,291,92,317,118,307,101)(16,121,300,95,318,115,306,104)(17,128,299,98,319,112,305,107)(18,125,298,91,320,119,304,110)(19,122,297,94,311,116,303,103)(20,129,296,97,312,113,302,106)(51,200,79,184,70,201,85,178)(52,197,78,187,61,208,84,171)(53,194,77,190,62,205,83,174)(54,191,76,183,63,202,82,177)(55,198,75,186,64,209,81,180)(56,195,74,189,65,206,90,173)(57,192,73,182,66,203,89,176)(58,199,72,185,67,210,88,179)(59,196,71,188,68,207,87,172)(60,193,80,181,69,204,86,175)(131,275,159,269,150,286,164,253)(132,272,158,262,141,283,163,256)(133,279,157,265,142,290,162,259)(134,276,156,268,143,287,161,252)(135,273,155,261,144,284,170,255)(136,280,154,264,145,281,169,258)(137,277,153,267,146,288,168,251)(138,274,152,270,147,285,167,254)(139,271,151,263,148,282,166,257)(140,278,160,266,149,289,165,260), (1,51,6,56)(2,52,7,57)(3,53,8,58)(4,54,9,59)(5,55,10,60)(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)(171,227,176,222)(172,228,177,223)(173,229,178,224)(174,230,179,225)(175,221,180,226)(181,215,186,220)(182,216,187,211)(183,217,188,212)(184,218,189,213)(185,219,190,214)(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)(251,302,256,307)(252,303,257,308)(253,304,258,309)(254,305,259,310)(255,306,260,301)(261,300,266,295)(262,291,267,296)(263,292,268,297)(264,293,269,298)(265,294,270,299)(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), (1,119,6,114)(2,118,7,113)(3,117,8,112)(4,116,9,111)(5,115,10,120)(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)(31,103,36,108)(32,102,37,107)(33,101,38,106)(34,110,39,105)(35,109,40,104)(41,100,46,95)(42,99,47,94)(43,98,48,93)(44,97,49,92)(45,96,50,91)(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)(71,143,76,148)(72,142,77,147)(73,141,78,146)(74,150,79,145)(75,149,80,144)(81,140,86,135)(82,139,87,134)(83,138,88,133)(84,137,89,132)(85,136,90,131)(171,272,176,277)(172,271,177,276)(173,280,178,275)(174,279,179,274)(175,278,180,273)(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)(201,253,206,258)(202,252,207,257)(203,251,208,256)(204,260,209,255)(205,259,210,254)(211,312,216,317)(212,311,217,316)(213,320,218,315)(214,319,219,314)(215,318,220,313)(231,308,236,303)(232,307,237,302)(233,306,238,301)(234,305,239,310)(235,304,240,309)(241,293,246,298)(242,292,247,297)(243,291,248,296)(244,300,249,295)(245,299,250,294), (1,240,34,224,25,241,50,218)(2,237,33,227,26,248,49,211)(3,234,32,230,27,245,48,214)(4,231,31,223,28,242,47,217)(5,238,40,226,29,249,46,220)(6,235,39,229,30,246,45,213)(7,232,38,222,21,243,44,216)(8,239,37,225,22,250,43,219)(9,236,36,228,23,247,42,212)(10,233,35,221,24,244,41,215)(11,126,295,100,313,120,301,109)(12,123,294,93,314,117,310,102)(13,130,293,96,315,114,309,105)(14,127,292,99,316,111,308,108)(15,124,291,92,317,118,307,101)(16,121,300,95,318,115,306,104)(17,128,299,98,319,112,305,107)(18,125,298,91,320,119,304,110)(19,122,297,94,311,116,303,103)(20,129,296,97,312,113,302,106)(51,200,79,184,70,201,85,178)(52,197,78,187,61,208,84,171)(53,194,77,190,62,205,83,174)(54,191,76,183,63,202,82,177)(55,198,75,186,64,209,81,180)(56,195,74,189,65,206,90,173)(57,192,73,182,66,203,89,176)(58,199,72,185,67,210,88,179)(59,196,71,188,68,207,87,172)(60,193,80,181,69,204,86,175)(131,275,159,269,150,286,164,253)(132,272,158,262,141,283,163,256)(133,279,157,265,142,290,162,259)(134,276,156,268,143,287,161,252)(135,273,155,261,144,284,170,255)(136,280,154,264,145,281,169,258)(137,277,153,267,146,288,168,251)(138,274,152,270,147,285,167,254)(139,271,151,263,148,282,166,257)(140,278,160,266,149,289,165,260), (1,51,6,56)(2,52,7,57)(3,53,8,58)(4,54,9,59)(5,55,10,60)(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)(171,227,176,222)(172,228,177,223)(173,229,178,224)(174,230,179,225)(175,221,180,226)(181,215,186,220)(182,216,187,211)(183,217,188,212)(184,218,189,213)(185,219,190,214)(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)(251,302,256,307)(252,303,257,308)(253,304,258,309)(254,305,259,310)(255,306,260,301)(261,300,266,295)(262,291,267,296)(263,292,268,297)(264,293,269,298)(265,294,270,299)(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)], [(1,119,6,114),(2,118,7,113),(3,117,8,112),(4,116,9,111),(5,115,10,120),(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),(31,103,36,108),(32,102,37,107),(33,101,38,106),(34,110,39,105),(35,109,40,104),(41,100,46,95),(42,99,47,94),(43,98,48,93),(44,97,49,92),(45,96,50,91),(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),(71,143,76,148),(72,142,77,147),(73,141,78,146),(74,150,79,145),(75,149,80,144),(81,140,86,135),(82,139,87,134),(83,138,88,133),(84,137,89,132),(85,136,90,131),(171,272,176,277),(172,271,177,276),(173,280,178,275),(174,279,179,274),(175,278,180,273),(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),(201,253,206,258),(202,252,207,257),(203,251,208,256),(204,260,209,255),(205,259,210,254),(211,312,216,317),(212,311,217,316),(213,320,218,315),(214,319,219,314),(215,318,220,313),(231,308,236,303),(232,307,237,302),(233,306,238,301),(234,305,239,310),(235,304,240,309),(241,293,246,298),(242,292,247,297),(243,291,248,296),(244,300,249,295),(245,299,250,294)], [(1,240,34,224,25,241,50,218),(2,237,33,227,26,248,49,211),(3,234,32,230,27,245,48,214),(4,231,31,223,28,242,47,217),(5,238,40,226,29,249,46,220),(6,235,39,229,30,246,45,213),(7,232,38,222,21,243,44,216),(8,239,37,225,22,250,43,219),(9,236,36,228,23,247,42,212),(10,233,35,221,24,244,41,215),(11,126,295,100,313,120,301,109),(12,123,294,93,314,117,310,102),(13,130,293,96,315,114,309,105),(14,127,292,99,316,111,308,108),(15,124,291,92,317,118,307,101),(16,121,300,95,318,115,306,104),(17,128,299,98,319,112,305,107),(18,125,298,91,320,119,304,110),(19,122,297,94,311,116,303,103),(20,129,296,97,312,113,302,106),(51,200,79,184,70,201,85,178),(52,197,78,187,61,208,84,171),(53,194,77,190,62,205,83,174),(54,191,76,183,63,202,82,177),(55,198,75,186,64,209,81,180),(56,195,74,189,65,206,90,173),(57,192,73,182,66,203,89,176),(58,199,72,185,67,210,88,179),(59,196,71,188,68,207,87,172),(60,193,80,181,69,204,86,175),(131,275,159,269,150,286,164,253),(132,272,158,262,141,283,163,256),(133,279,157,265,142,290,162,259),(134,276,156,268,143,287,161,252),(135,273,155,261,144,284,170,255),(136,280,154,264,145,281,169,258),(137,277,153,267,146,288,168,251),(138,274,152,270,147,285,167,254),(139,271,151,263,148,282,166,257),(140,278,160,266,149,289,165,260)], [(1,51,6,56),(2,52,7,57),(3,53,8,58),(4,54,9,59),(5,55,10,60),(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),(171,227,176,222),(172,228,177,223),(173,229,178,224),(174,230,179,225),(175,221,180,226),(181,215,186,220),(182,216,187,211),(183,217,188,212),(184,218,189,213),(185,219,190,214),(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),(251,302,256,307),(252,303,257,308),(253,304,258,309),(254,305,259,310),(255,306,260,301),(261,300,266,295),(262,291,267,296),(263,292,268,297),(264,293,269,298),(265,294,270,299),(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⋊C8Dic5⋊3Q8C10.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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