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G = Dic5.5M4(2)  order 320 = 26·5

2nd non-split extension by Dic5 of M4(2) acting via M4(2)/C8=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.27D10, Dic5.5M4(2), C52C89Q8, C4⋊C8.11D5, C56(C84Q8), C4.51(Q8×D5), C10.25(C4×Q8), C408C4.9C2, (C2×C8).212D10, C4⋊Dic5.23C4, C20.109(C2×Q8), C10.35(C8○D4), (C4×C20).54C22, (C8×Dic5).17C2, (C4×Dic10).7C2, C2.15(D5×M4(2)), C20.8Q8.7C2, C20.301(C4○D4), (C2×C20).826C23, (C2×C40).205C22, (C2×Dic10).21C4, C10.D4.23C4, C10.58(C2×M4(2)), C4.127(D42D5), C42.D5.2C2, C2.7(Dic53Q8), C2.13(D20.3C4), (C4×Dic5).304C22, (C5×C4⋊C8).16C2, (C2×C4).34(C4×D5), C22.108(C2×C4×D5), (C2×C20).215(C2×C4), (C2×Dic5).22(C2×C4), (C2×C4).768(C22×D5), (C2×C10).182(C22×C4), (C2×C52C8).197C22, SmallGroup(320,455)

Series: Derived Chief Lower central Upper central

C1C2×C10 — Dic5.5M4(2)
C1C5C10C20C2×C20C4×Dic5C4×Dic10 — Dic5.5M4(2)
C5C2×C10 — Dic5.5M4(2)
C1C2×C4C4⋊C8

Generators and relations for Dic5.5M4(2)
 G = < a,b,c,d | a10=c8=1, b2=d2=a5, bab-1=a-1, ac=ca, ad=da, cbc-1=dbd-1=a5b, dcd-1=a5c5 >

Subgroups: 254 in 94 conjugacy classes, 51 normal (47 characteristic)
C1, C2 [×3], C4 [×2], C4 [×7], C22, C5, C8 [×5], C2×C4 [×3], C2×C4 [×4], Q8 [×2], C10 [×3], C42, C42 [×2], C4⋊C4 [×3], C2×C8 [×2], C2×C8 [×2], C2×Q8, Dic5 [×2], Dic5 [×3], C20 [×2], C20 [×2], C2×C10, C4×C8, C8⋊C4 [×2], C4⋊C8, C4⋊C8 [×2], C4×Q8, C52C8 [×2], C52C8, C40 [×2], Dic10 [×2], C2×Dic5 [×4], C2×C20 [×3], C84Q8, C2×C52C8 [×2], C4×Dic5 [×2], C10.D4 [×2], C4⋊Dic5, C4×C20, C2×C40 [×2], C2×Dic10, C42.D5, C8×Dic5, C20.8Q8 [×2], C408C4, C5×C4⋊C8, C4×Dic10, Dic5.5M4(2)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], Q8 [×2], C23, D5, M4(2) [×2], C22×C4, C2×Q8, C4○D4, D10 [×3], C4×Q8, C2×M4(2), C8○D4, C4×D5 [×2], C22×D5, C84Q8, C2×C4×D5, D42D5, Q8×D5, Dic53Q8, D20.3C4, D5×M4(2), Dic5.5M4(2)

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

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

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

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

68 conjugacy classes

class 1 2A2B2C4A4B4C4D4E4F4G4H4I4J4K4L5A5B8A8B8C8D8E8F8G8H8I8J8K8L10A···10F20A···20H20I···20P40A···40P
order12224444444444445588888888888810···1020···2020···2040···40
size1111111144101010102020222222441010101020202···22···24···44···4

68 irreducible representations

dim1111111111222222222444
type+++++++-+++--
imageC1C2C2C2C2C2C2C4C4C4Q8D5M4(2)C4○D4D10D10C8○D4C4×D5D20.3C4D42D5Q8×D5D5×M4(2)
kernelDic5.5M4(2)C42.D5C8×Dic5C20.8Q8C408C4C5×C4⋊C8C4×Dic10C10.D4C4⋊Dic5C2×Dic10C52C8C4⋊C8Dic5C20C42C2×C8C10C2×C4C2C4C4C2
# reps11121114222242244816224

Matrix representation of Dic5.5M4(2) in GL4(𝔽41) generated by

04000
1700
00400
00040
,
292500
271200
003221
0009
,
332700
14800
00328
002638
,
11900
323000
002713
002914
G:=sub<GL(4,GF(41))| [0,1,0,0,40,7,0,0,0,0,40,0,0,0,0,40],[29,27,0,0,25,12,0,0,0,0,32,0,0,0,21,9],[33,14,0,0,27,8,0,0,0,0,3,26,0,0,28,38],[11,32,0,0,9,30,0,0,0,0,27,29,0,0,13,14] >;

Dic5.5M4(2) in GAP, Magma, Sage, TeX

{\rm Dic}_5._5M_4(2)
% in TeX

G:=Group("Dic5.5M4(2)");
// GroupNames label

G:=SmallGroup(320,455);
// by ID

G=gap.SmallGroup(320,455);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,701,120,219,58,136,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^10=c^8=1,b^2=d^2=a^5,b*a*b^-1=a^-1,a*c=c*a,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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