metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.198D10, C20.23M4(2), C4⋊C8.12D5, C5⋊2C8⋊10Q8, C5⋊7(C8⋊4Q8), C4.53(Q8×D5), C10.27(C4×Q8), (C2×C8).180D10, C4.3(C8⋊D5), C4⋊Dic5.29C4, C20.111(C2×Q8), C40⋊8C4.10C2, C10.51(C8○D4), (C4×C20).57C22, C20.8Q8.9C2, C20.303(C4○D4), (C2×C20).828C23, (C2×C40).207C22, (C2×Dic10).26C4, (C4×Dic10).10C2, C10.D4.22C4, C10.41(C2×M4(2)), C4.129(D4⋊2D5), C2.8(Dic5⋊3Q8), C2.12(D20.2C4), (C4×Dic5).204C22, (C5×C4⋊C8).18C2, (C4×C5⋊2C8).6C2, (C2×C4).71(C4×D5), C2.11(C2×C8⋊D5), C22.109(C2×C4×D5), (C2×C20).330(C2×C4), (C2×Dic5).23(C2×C4), (C2×C4).770(C22×D5), (C2×C10).184(C22×C4), (C2×C5⋊2C8).311C22, SmallGroup(320,458)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.198D10
G = < a,b,c,d | a4=b4=1, c10=a2b, d2=a2, ab=ba, cac-1=dad-1=a-1, bc=cb, bd=db, dcd-1=b2c9 >
Subgroups: 254 in 94 conjugacy classes, 53 normal (31 characteristic)
C1, C2, C4, C4, C4, C22, C5, C8, C2×C4, C2×C4, Q8, C10, C42, C42, C4⋊C4, C2×C8, C2×C8, C2×Q8, Dic5, C20, C20, C20, C2×C10, C4×C8, C8⋊C4, C4⋊C8, C4⋊C8, C4×Q8, C5⋊2C8, C5⋊2C8, C40, Dic10, C2×Dic5, C2×C20, C8⋊4Q8, C2×C5⋊2C8, C4×Dic5, C10.D4, C4⋊Dic5, C4×C20, C2×C40, C2×Dic10, C4×C5⋊2C8, C20.8Q8, C40⋊8C4, C5×C4⋊C8, C4×Dic10, C42.198D10
Quotients: C1, C2, C4, C22, C2×C4, Q8, C23, D5, M4(2), C22×C4, C2×Q8, C4○D4, D10, C4×Q8, C2×M4(2), C8○D4, C4×D5, C22×D5, C8⋊4Q8, C8⋊D5, C2×C4×D5, D4⋊2D5, Q8×D5, Dic5⋊3Q8, C2×C8⋊D5, D20.2C4, C42.198D10
►Regular action on 320 points
Generators in S320
(1 140 75 243)(2 244 76 141)(3 142 77 245)(4 246 78 143)(5 144 79 247)(6 248 80 145)(7 146 41 249)(8 250 42 147)(9 148 43 251)(10 252 44 149)(11 150 45 253)(12 254 46 151)(13 152 47 255)(14 256 48 153)(15 154 49 257)(16 258 50 155)(17 156 51 259)(18 260 52 157)(19 158 53 261)(20 262 54 159)(21 160 55 263)(22 264 56 121)(23 122 57 265)(24 266 58 123)(25 124 59 267)(26 268 60 125)(27 126 61 269)(28 270 62 127)(29 128 63 271)(30 272 64 129)(31 130 65 273)(32 274 66 131)(33 132 67 275)(34 276 68 133)(35 134 69 277)(36 278 70 135)(37 136 71 279)(38 280 72 137)(39 138 73 241)(40 242 74 139)(81 281 215 192)(82 193 216 282)(83 283 217 194)(84 195 218 284)(85 285 219 196)(86 197 220 286)(87 287 221 198)(88 199 222 288)(89 289 223 200)(90 161 224 290)(91 291 225 162)(92 163 226 292)(93 293 227 164)(94 165 228 294)(95 295 229 166)(96 167 230 296)(97 297 231 168)(98 169 232 298)(99 299 233 170)(100 171 234 300)(101 301 235 172)(102 173 236 302)(103 303 237 174)(104 175 238 304)(105 305 239 176)(106 177 240 306)(107 307 201 178)(108 179 202 308)(109 309 203 180)(110 181 204 310)(111 311 205 182)(112 183 206 312)(113 313 207 184)(114 185 208 314)(115 315 209 186)(116 187 210 316)(117 317 211 188)(118 189 212 318)(119 319 213 190)(120 191 214 320)
(1 45 21 65)(2 46 22 66)(3 47 23 67)(4 48 24 68)(5 49 25 69)(6 50 26 70)(7 51 27 71)(8 52 28 72)(9 53 29 73)(10 54 30 74)(11 55 31 75)(12 56 32 76)(13 57 33 77)(14 58 34 78)(15 59 35 79)(16 60 36 80)(17 61 37 41)(18 62 38 42)(19 63 39 43)(20 64 40 44)(81 225 101 205)(82 226 102 206)(83 227 103 207)(84 228 104 208)(85 229 105 209)(86 230 106 210)(87 231 107 211)(88 232 108 212)(89 233 109 213)(90 234 110 214)(91 235 111 215)(92 236 112 216)(93 237 113 217)(94 238 114 218)(95 239 115 219)(96 240 116 220)(97 201 117 221)(98 202 118 222)(99 203 119 223)(100 204 120 224)(121 274 141 254)(122 275 142 255)(123 276 143 256)(124 277 144 257)(125 278 145 258)(126 279 146 259)(127 280 147 260)(128 241 148 261)(129 242 149 262)(130 243 150 263)(131 244 151 264)(132 245 152 265)(133 246 153 266)(134 247 154 267)(135 248 155 268)(136 249 156 269)(137 250 157 270)(138 251 158 271)(139 252 159 272)(140 253 160 273)(161 300 181 320)(162 301 182 281)(163 302 183 282)(164 303 184 283)(165 304 185 284)(166 305 186 285)(167 306 187 286)(168 307 188 287)(169 308 189 288)(170 309 190 289)(171 310 191 290)(172 311 192 291)(173 312 193 292)(174 313 194 293)(175 314 195 294)(176 315 196 295)(177 316 197 296)(178 317 198 297)(179 318 199 298)(180 319 200 299)
(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)
(1 105 75 239)(2 94 76 228)(3 83 77 217)(4 112 78 206)(5 101 79 235)(6 90 80 224)(7 119 41 213)(8 108 42 202)(9 97 43 231)(10 86 44 220)(11 115 45 209)(12 104 46 238)(13 93 47 227)(14 82 48 216)(15 111 49 205)(16 100 50 234)(17 89 51 223)(18 118 52 212)(19 107 53 201)(20 96 54 230)(21 85 55 219)(22 114 56 208)(23 103 57 237)(24 92 58 226)(25 81 59 215)(26 110 60 204)(27 99 61 233)(28 88 62 222)(29 117 63 211)(30 106 64 240)(31 95 65 229)(32 84 66 218)(33 113 67 207)(34 102 68 236)(35 91 69 225)(36 120 70 214)(37 109 71 203)(38 98 72 232)(39 87 73 221)(40 116 74 210)(121 185 264 314)(122 174 265 303)(123 163 266 292)(124 192 267 281)(125 181 268 310)(126 170 269 299)(127 199 270 288)(128 188 271 317)(129 177 272 306)(130 166 273 295)(131 195 274 284)(132 184 275 313)(133 173 276 302)(134 162 277 291)(135 191 278 320)(136 180 279 309)(137 169 280 298)(138 198 241 287)(139 187 242 316)(140 176 243 305)(141 165 244 294)(142 194 245 283)(143 183 246 312)(144 172 247 301)(145 161 248 290)(146 190 249 319)(147 179 250 308)(148 168 251 297)(149 197 252 286)(150 186 253 315)(151 175 254 304)(152 164 255 293)(153 193 256 282)(154 182 257 311)(155 171 258 300)(156 200 259 289)(157 189 260 318)(158 178 261 307)(159 167 262 296)(160 196 263 285)
G:=sub<Sym(320)| (1,140,75,243)(2,244,76,141)(3,142,77,245)(4,246,78,143)(5,144,79,247)(6,248,80,145)(7,146,41,249)(8,250,42,147)(9,148,43,251)(10,252,44,149)(11,150,45,253)(12,254,46,151)(13,152,47,255)(14,256,48,153)(15,154,49,257)(16,258,50,155)(17,156,51,259)(18,260,52,157)(19,158,53,261)(20,262,54,159)(21,160,55,263)(22,264,56,121)(23,122,57,265)(24,266,58,123)(25,124,59,267)(26,268,60,125)(27,126,61,269)(28,270,62,127)(29,128,63,271)(30,272,64,129)(31,130,65,273)(32,274,66,131)(33,132,67,275)(34,276,68,133)(35,134,69,277)(36,278,70,135)(37,136,71,279)(38,280,72,137)(39,138,73,241)(40,242,74,139)(81,281,215,192)(82,193,216,282)(83,283,217,194)(84,195,218,284)(85,285,219,196)(86,197,220,286)(87,287,221,198)(88,199,222,288)(89,289,223,200)(90,161,224,290)(91,291,225,162)(92,163,226,292)(93,293,227,164)(94,165,228,294)(95,295,229,166)(96,167,230,296)(97,297,231,168)(98,169,232,298)(99,299,233,170)(100,171,234,300)(101,301,235,172)(102,173,236,302)(103,303,237,174)(104,175,238,304)(105,305,239,176)(106,177,240,306)(107,307,201,178)(108,179,202,308)(109,309,203,180)(110,181,204,310)(111,311,205,182)(112,183,206,312)(113,313,207,184)(114,185,208,314)(115,315,209,186)(116,187,210,316)(117,317,211,188)(118,189,212,318)(119,319,213,190)(120,191,214,320), (1,45,21,65)(2,46,22,66)(3,47,23,67)(4,48,24,68)(5,49,25,69)(6,50,26,70)(7,51,27,71)(8,52,28,72)(9,53,29,73)(10,54,30,74)(11,55,31,75)(12,56,32,76)(13,57,33,77)(14,58,34,78)(15,59,35,79)(16,60,36,80)(17,61,37,41)(18,62,38,42)(19,63,39,43)(20,64,40,44)(81,225,101,205)(82,226,102,206)(83,227,103,207)(84,228,104,208)(85,229,105,209)(86,230,106,210)(87,231,107,211)(88,232,108,212)(89,233,109,213)(90,234,110,214)(91,235,111,215)(92,236,112,216)(93,237,113,217)(94,238,114,218)(95,239,115,219)(96,240,116,220)(97,201,117,221)(98,202,118,222)(99,203,119,223)(100,204,120,224)(121,274,141,254)(122,275,142,255)(123,276,143,256)(124,277,144,257)(125,278,145,258)(126,279,146,259)(127,280,147,260)(128,241,148,261)(129,242,149,262)(130,243,150,263)(131,244,151,264)(132,245,152,265)(133,246,153,266)(134,247,154,267)(135,248,155,268)(136,249,156,269)(137,250,157,270)(138,251,158,271)(139,252,159,272)(140,253,160,273)(161,300,181,320)(162,301,182,281)(163,302,183,282)(164,303,184,283)(165,304,185,284)(166,305,186,285)(167,306,187,286)(168,307,188,287)(169,308,189,288)(170,309,190,289)(171,310,191,290)(172,311,192,291)(173,312,193,292)(174,313,194,293)(175,314,195,294)(176,315,196,295)(177,316,197,296)(178,317,198,297)(179,318,199,298)(180,319,200,299), (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), (1,105,75,239)(2,94,76,228)(3,83,77,217)(4,112,78,206)(5,101,79,235)(6,90,80,224)(7,119,41,213)(8,108,42,202)(9,97,43,231)(10,86,44,220)(11,115,45,209)(12,104,46,238)(13,93,47,227)(14,82,48,216)(15,111,49,205)(16,100,50,234)(17,89,51,223)(18,118,52,212)(19,107,53,201)(20,96,54,230)(21,85,55,219)(22,114,56,208)(23,103,57,237)(24,92,58,226)(25,81,59,215)(26,110,60,204)(27,99,61,233)(28,88,62,222)(29,117,63,211)(30,106,64,240)(31,95,65,229)(32,84,66,218)(33,113,67,207)(34,102,68,236)(35,91,69,225)(36,120,70,214)(37,109,71,203)(38,98,72,232)(39,87,73,221)(40,116,74,210)(121,185,264,314)(122,174,265,303)(123,163,266,292)(124,192,267,281)(125,181,268,310)(126,170,269,299)(127,199,270,288)(128,188,271,317)(129,177,272,306)(130,166,273,295)(131,195,274,284)(132,184,275,313)(133,173,276,302)(134,162,277,291)(135,191,278,320)(136,180,279,309)(137,169,280,298)(138,198,241,287)(139,187,242,316)(140,176,243,305)(141,165,244,294)(142,194,245,283)(143,183,246,312)(144,172,247,301)(145,161,248,290)(146,190,249,319)(147,179,250,308)(148,168,251,297)(149,197,252,286)(150,186,253,315)(151,175,254,304)(152,164,255,293)(153,193,256,282)(154,182,257,311)(155,171,258,300)(156,200,259,289)(157,189,260,318)(158,178,261,307)(159,167,262,296)(160,196,263,285)>;
G:=Group( (1,140,75,243)(2,244,76,141)(3,142,77,245)(4,246,78,143)(5,144,79,247)(6,248,80,145)(7,146,41,249)(8,250,42,147)(9,148,43,251)(10,252,44,149)(11,150,45,253)(12,254,46,151)(13,152,47,255)(14,256,48,153)(15,154,49,257)(16,258,50,155)(17,156,51,259)(18,260,52,157)(19,158,53,261)(20,262,54,159)(21,160,55,263)(22,264,56,121)(23,122,57,265)(24,266,58,123)(25,124,59,267)(26,268,60,125)(27,126,61,269)(28,270,62,127)(29,128,63,271)(30,272,64,129)(31,130,65,273)(32,274,66,131)(33,132,67,275)(34,276,68,133)(35,134,69,277)(36,278,70,135)(37,136,71,279)(38,280,72,137)(39,138,73,241)(40,242,74,139)(81,281,215,192)(82,193,216,282)(83,283,217,194)(84,195,218,284)(85,285,219,196)(86,197,220,286)(87,287,221,198)(88,199,222,288)(89,289,223,200)(90,161,224,290)(91,291,225,162)(92,163,226,292)(93,293,227,164)(94,165,228,294)(95,295,229,166)(96,167,230,296)(97,297,231,168)(98,169,232,298)(99,299,233,170)(100,171,234,300)(101,301,235,172)(102,173,236,302)(103,303,237,174)(104,175,238,304)(105,305,239,176)(106,177,240,306)(107,307,201,178)(108,179,202,308)(109,309,203,180)(110,181,204,310)(111,311,205,182)(112,183,206,312)(113,313,207,184)(114,185,208,314)(115,315,209,186)(116,187,210,316)(117,317,211,188)(118,189,212,318)(119,319,213,190)(120,191,214,320), (1,45,21,65)(2,46,22,66)(3,47,23,67)(4,48,24,68)(5,49,25,69)(6,50,26,70)(7,51,27,71)(8,52,28,72)(9,53,29,73)(10,54,30,74)(11,55,31,75)(12,56,32,76)(13,57,33,77)(14,58,34,78)(15,59,35,79)(16,60,36,80)(17,61,37,41)(18,62,38,42)(19,63,39,43)(20,64,40,44)(81,225,101,205)(82,226,102,206)(83,227,103,207)(84,228,104,208)(85,229,105,209)(86,230,106,210)(87,231,107,211)(88,232,108,212)(89,233,109,213)(90,234,110,214)(91,235,111,215)(92,236,112,216)(93,237,113,217)(94,238,114,218)(95,239,115,219)(96,240,116,220)(97,201,117,221)(98,202,118,222)(99,203,119,223)(100,204,120,224)(121,274,141,254)(122,275,142,255)(123,276,143,256)(124,277,144,257)(125,278,145,258)(126,279,146,259)(127,280,147,260)(128,241,148,261)(129,242,149,262)(130,243,150,263)(131,244,151,264)(132,245,152,265)(133,246,153,266)(134,247,154,267)(135,248,155,268)(136,249,156,269)(137,250,157,270)(138,251,158,271)(139,252,159,272)(140,253,160,273)(161,300,181,320)(162,301,182,281)(163,302,183,282)(164,303,184,283)(165,304,185,284)(166,305,186,285)(167,306,187,286)(168,307,188,287)(169,308,189,288)(170,309,190,289)(171,310,191,290)(172,311,192,291)(173,312,193,292)(174,313,194,293)(175,314,195,294)(176,315,196,295)(177,316,197,296)(178,317,198,297)(179,318,199,298)(180,319,200,299), (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), (1,105,75,239)(2,94,76,228)(3,83,77,217)(4,112,78,206)(5,101,79,235)(6,90,80,224)(7,119,41,213)(8,108,42,202)(9,97,43,231)(10,86,44,220)(11,115,45,209)(12,104,46,238)(13,93,47,227)(14,82,48,216)(15,111,49,205)(16,100,50,234)(17,89,51,223)(18,118,52,212)(19,107,53,201)(20,96,54,230)(21,85,55,219)(22,114,56,208)(23,103,57,237)(24,92,58,226)(25,81,59,215)(26,110,60,204)(27,99,61,233)(28,88,62,222)(29,117,63,211)(30,106,64,240)(31,95,65,229)(32,84,66,218)(33,113,67,207)(34,102,68,236)(35,91,69,225)(36,120,70,214)(37,109,71,203)(38,98,72,232)(39,87,73,221)(40,116,74,210)(121,185,264,314)(122,174,265,303)(123,163,266,292)(124,192,267,281)(125,181,268,310)(126,170,269,299)(127,199,270,288)(128,188,271,317)(129,177,272,306)(130,166,273,295)(131,195,274,284)(132,184,275,313)(133,173,276,302)(134,162,277,291)(135,191,278,320)(136,180,279,309)(137,169,280,298)(138,198,241,287)(139,187,242,316)(140,176,243,305)(141,165,244,294)(142,194,245,283)(143,183,246,312)(144,172,247,301)(145,161,248,290)(146,190,249,319)(147,179,250,308)(148,168,251,297)(149,197,252,286)(150,186,253,315)(151,175,254,304)(152,164,255,293)(153,193,256,282)(154,182,257,311)(155,171,258,300)(156,200,259,289)(157,189,260,318)(158,178,261,307)(159,167,262,296)(160,196,263,285) );
G=PermutationGroup([[(1,140,75,243),(2,244,76,141),(3,142,77,245),(4,246,78,143),(5,144,79,247),(6,248,80,145),(7,146,41,249),(8,250,42,147),(9,148,43,251),(10,252,44,149),(11,150,45,253),(12,254,46,151),(13,152,47,255),(14,256,48,153),(15,154,49,257),(16,258,50,155),(17,156,51,259),(18,260,52,157),(19,158,53,261),(20,262,54,159),(21,160,55,263),(22,264,56,121),(23,122,57,265),(24,266,58,123),(25,124,59,267),(26,268,60,125),(27,126,61,269),(28,270,62,127),(29,128,63,271),(30,272,64,129),(31,130,65,273),(32,274,66,131),(33,132,67,275),(34,276,68,133),(35,134,69,277),(36,278,70,135),(37,136,71,279),(38,280,72,137),(39,138,73,241),(40,242,74,139),(81,281,215,192),(82,193,216,282),(83,283,217,194),(84,195,218,284),(85,285,219,196),(86,197,220,286),(87,287,221,198),(88,199,222,288),(89,289,223,200),(90,161,224,290),(91,291,225,162),(92,163,226,292),(93,293,227,164),(94,165,228,294),(95,295,229,166),(96,167,230,296),(97,297,231,168),(98,169,232,298),(99,299,233,170),(100,171,234,300),(101,301,235,172),(102,173,236,302),(103,303,237,174),(104,175,238,304),(105,305,239,176),(106,177,240,306),(107,307,201,178),(108,179,202,308),(109,309,203,180),(110,181,204,310),(111,311,205,182),(112,183,206,312),(113,313,207,184),(114,185,208,314),(115,315,209,186),(116,187,210,316),(117,317,211,188),(118,189,212,318),(119,319,213,190),(120,191,214,320)], [(1,45,21,65),(2,46,22,66),(3,47,23,67),(4,48,24,68),(5,49,25,69),(6,50,26,70),(7,51,27,71),(8,52,28,72),(9,53,29,73),(10,54,30,74),(11,55,31,75),(12,56,32,76),(13,57,33,77),(14,58,34,78),(15,59,35,79),(16,60,36,80),(17,61,37,41),(18,62,38,42),(19,63,39,43),(20,64,40,44),(81,225,101,205),(82,226,102,206),(83,227,103,207),(84,228,104,208),(85,229,105,209),(86,230,106,210),(87,231,107,211),(88,232,108,212),(89,233,109,213),(90,234,110,214),(91,235,111,215),(92,236,112,216),(93,237,113,217),(94,238,114,218),(95,239,115,219),(96,240,116,220),(97,201,117,221),(98,202,118,222),(99,203,119,223),(100,204,120,224),(121,274,141,254),(122,275,142,255),(123,276,143,256),(124,277,144,257),(125,278,145,258),(126,279,146,259),(127,280,147,260),(128,241,148,261),(129,242,149,262),(130,243,150,263),(131,244,151,264),(132,245,152,265),(133,246,153,266),(134,247,154,267),(135,248,155,268),(136,249,156,269),(137,250,157,270),(138,251,158,271),(139,252,159,272),(140,253,160,273),(161,300,181,320),(162,301,182,281),(163,302,183,282),(164,303,184,283),(165,304,185,284),(166,305,186,285),(167,306,187,286),(168,307,188,287),(169,308,189,288),(170,309,190,289),(171,310,191,290),(172,311,192,291),(173,312,193,292),(174,313,194,293),(175,314,195,294),(176,315,196,295),(177,316,197,296),(178,317,198,297),(179,318,199,298),(180,319,200,299)], [(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)], [(1,105,75,239),(2,94,76,228),(3,83,77,217),(4,112,78,206),(5,101,79,235),(6,90,80,224),(7,119,41,213),(8,108,42,202),(9,97,43,231),(10,86,44,220),(11,115,45,209),(12,104,46,238),(13,93,47,227),(14,82,48,216),(15,111,49,205),(16,100,50,234),(17,89,51,223),(18,118,52,212),(19,107,53,201),(20,96,54,230),(21,85,55,219),(22,114,56,208),(23,103,57,237),(24,92,58,226),(25,81,59,215),(26,110,60,204),(27,99,61,233),(28,88,62,222),(29,117,63,211),(30,106,64,240),(31,95,65,229),(32,84,66,218),(33,113,67,207),(34,102,68,236),(35,91,69,225),(36,120,70,214),(37,109,71,203),(38,98,72,232),(39,87,73,221),(40,116,74,210),(121,185,264,314),(122,174,265,303),(123,163,266,292),(124,192,267,281),(125,181,268,310),(126,170,269,299),(127,199,270,288),(128,188,271,317),(129,177,272,306),(130,166,273,295),(131,195,274,284),(132,184,275,313),(133,173,276,302),(134,162,277,291),(135,191,278,320),(136,180,279,309),(137,169,280,298),(138,198,241,287),(139,187,242,316),(140,176,243,305),(141,165,244,294),(142,194,245,283),(143,183,246,312),(144,172,247,301),(145,161,248,290),(146,190,249,319),(147,179,250,308),(148,168,251,297),(149,197,252,286),(150,186,253,315),(151,175,254,304),(152,164,255,293),(153,193,256,282),(154,182,257,311),(155,171,258,300),(156,200,259,289),(157,189,260,318),(158,178,261,307),(159,167,262,296),(160,196,263,285)]])
68 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 5A | 5B | 8A | 8B | 8C | 8D | 8E | ··· | 8L | 10A | ··· | 10F | 20A | ··· | 20H | 20I | ··· | 20P | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | 2 | 4 | 4 | 4 | 4 | 10 | ··· | 10 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | + | + | - | - | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | Q8 | D5 | M4(2) | C4○D4 | D10 | D10 | C8○D4 | C4×D5 | C8⋊D5 | D4⋊2D5 | Q8×D5 | D20.2C4 |
| kernel | C42.198D10 | C4×C5⋊2C8 | C20.8Q8 | C40⋊8C4 | C5×C4⋊C8 | C4×Dic10 | C10.D4 | C4⋊Dic5 | C2×Dic10 | C5⋊2C8 | C4⋊C8 | C20 | C20 | C42 | C2×C8 | C10 | C2×C4 | C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 4 | 2 | 2 | 2 | 2 | 4 | 2 | 2 | 4 | 4 | 8 | 16 | 2 | 2 | 4 |
Matrix representation of C42.198D10 ►in GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 7 | 38 |
| 0 | 0 | 3 | 34 |
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 10 | 9 | 0 | 0 |
| 32 | 32 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 6 | 38 | 0 | 0 |
| 39 | 35 | 0 | 0 |
| 0 | 0 | 0 | 9 |
| 0 | 0 | 9 | 0 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,7,3,0,0,38,34],[9,0,0,0,0,9,0,0,0,0,40,0,0,0,0,40],[10,32,0,0,9,32,0,0,0,0,0,1,0,0,1,0],[6,39,0,0,38,35,0,0,0,0,0,9,0,0,9,0] >;
C42.198D10 in GAP, Magma, Sage, TeX
C_4^2._{198}D_{10} % in TeX
G:=Group("C4^2.198D10"); // GroupNames label
G:=SmallGroup(320,458);
// by ID
G=gap.SmallGroup(320,458);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,64,758,135,142,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=a^2*b,d^2=a^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,b*c=c*b,b*d=d*b,d*c*d^-1=b^2*c^9>;
// generators/relations