metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20.31C42, C4⋊C4⋊3Dic5, (C2×C20).4Q8, C4⋊Dic5⋊17C4, (C2×C10).35D8, C20.37(C4⋊C4), C4.1(C4×Dic5), (C2×C20).102D4, (C2×C4).125D20, C4.1(C4⋊Dic5), (C2×C10).14Q16, C10.9(C4.Q8), C5⋊3(C22.4Q16), (C2×C10).38SD16, (C2×C4).23Dic10, C10.12(C2.D8), C22.9(Q8⋊D5), C2.2(D20⋊6C4), C20.54(C22⋊C4), C22.16(D4⋊D5), C2.2(C10.Q16), C2.1(Q8⋊Dic5), C4.6(C10.D4), C2.1(D4⋊Dic5), C2.2(C10.D8), (C22×C10).178D4, (C22×C4).323D10, C23.94(C5⋊D4), C22.9(D4.D5), C2.2(C20.Q8), C4.31(D10⋊C4), C10.36(D4⋊C4), C22.6(C5⋊Q16), C10.11(Q8⋊C4), (C22×C20).118C22, C22.26(C23.D5), C22.38(D10⋊C4), C2.6(C10.10C42), C10.24(C2.C42), C22.20(C10.D4), (C5×C4⋊C4)⋊12C4, (C2×C5⋊2C8)⋊6C4, (C2×C4⋊C4).2D5, (C10×C4⋊C4).1C2, (C2×C4).138(C4×D5), (C2×C10).63(C4⋊C4), (C2×C20).227(C2×C4), (C2×C4⋊Dic5).28C2, (C22×C5⋊2C8).1C2, (C2×C4).35(C2×Dic5), (C2×C4).174(C5⋊D4), (C2×C10).151(C22⋊C4), SmallGroup(320,87)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C20.31C42
G = < a,b,c | a20=c4=1, b4=a10, bab-1=a9, cac-1=a11, cbc-1=a15b >
Subgroups: 342 in 114 conjugacy classes, 67 normal (59 characteristic)
C1, C2, C4, C4, C22, C5, C8, C2×C4, C2×C4, C23, C10, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, Dic5, C20, C20, C2×C10, C2×C4⋊C4, C2×C4⋊C4, C22×C8, C5⋊2C8, C2×Dic5, C2×C20, C2×C20, C22×C10, C22.4Q16, C2×C5⋊2C8, C2×C5⋊2C8, C4⋊Dic5, C4⋊Dic5, C5×C4⋊C4, C5×C4⋊C4, C22×Dic5, C22×C20, C22×C20, C22×C5⋊2C8, C2×C4⋊Dic5, C10×C4⋊C4, C20.31C42
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, D5, C42, C22⋊C4, C4⋊C4, D8, SD16, Q16, Dic5, D10, C2.C42, D4⋊C4, Q8⋊C4, C4.Q8, C2.D8, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C22.4Q16, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, D4⋊D5, D4.D5, Q8⋊D5, C5⋊Q16, C23.D5, C10.D8, C20.Q8, D20⋊6C4, C10.Q16, C10.10C42, D4⋊Dic5, Q8⋊Dic5, C20.31C42
►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)
(1 200 175 279 11 190 165 269)(2 189 176 268 12 199 166 278)(3 198 177 277 13 188 167 267)(4 187 178 266 14 197 168 276)(5 196 179 275 15 186 169 265)(6 185 180 264 16 195 170 274)(7 194 161 273 17 184 171 263)(8 183 162 262 18 193 172 272)(9 192 163 271 19 182 173 261)(10 181 164 280 20 191 174 270)(21 288 308 143 31 298 318 153)(22 297 309 152 32 287 319 142)(23 286 310 141 33 296 320 151)(24 295 311 150 34 285 301 160)(25 284 312 159 35 294 302 149)(26 293 313 148 36 283 303 158)(27 282 314 157 37 292 304 147)(28 291 315 146 38 281 305 156)(29 300 316 155 39 290 306 145)(30 289 317 144 40 299 307 154)(41 204 250 239 51 214 260 229)(42 213 251 228 52 203 241 238)(43 202 252 237 53 212 242 227)(44 211 253 226 54 201 243 236)(45 220 254 235 55 210 244 225)(46 209 255 224 56 219 245 234)(47 218 256 233 57 208 246 223)(48 207 257 222 58 217 247 232)(49 216 258 231 59 206 248 221)(50 205 259 240 60 215 249 230)(61 127 112 94 71 137 102 84)(62 136 113 83 72 126 103 93)(63 125 114 92 73 135 104 82)(64 134 115 81 74 124 105 91)(65 123 116 90 75 133 106 100)(66 132 117 99 76 122 107 89)(67 121 118 88 77 131 108 98)(68 130 119 97 78 140 109 87)(69 139 120 86 79 129 110 96)(70 128 101 95 80 138 111 85)
(1 251 293 63)(2 242 294 74)(3 253 295 65)(4 244 296 76)(5 255 297 67)(6 246 298 78)(7 257 299 69)(8 248 300 80)(9 259 281 71)(10 250 282 62)(11 241 283 73)(12 252 284 64)(13 243 285 75)(14 254 286 66)(15 245 287 77)(16 256 288 68)(17 247 289 79)(18 258 290 70)(19 249 291 61)(20 260 292 72)(21 92 264 203)(22 83 265 214)(23 94 266 205)(24 85 267 216)(25 96 268 207)(26 87 269 218)(27 98 270 209)(28 89 271 220)(29 100 272 211)(30 91 273 202)(31 82 274 213)(32 93 275 204)(33 84 276 215)(34 95 277 206)(35 86 278 217)(36 97 279 208)(37 88 280 219)(38 99 261 210)(39 90 262 201)(40 81 263 212)(41 157 103 164)(42 148 104 175)(43 159 105 166)(44 150 106 177)(45 141 107 168)(46 152 108 179)(47 143 109 170)(48 154 110 161)(49 145 111 172)(50 156 112 163)(51 147 113 174)(52 158 114 165)(53 149 115 176)(54 160 116 167)(55 151 117 178)(56 142 118 169)(57 153 119 180)(58 144 120 171)(59 155 101 162)(60 146 102 173)(121 191 224 304)(122 182 225 315)(123 193 226 306)(124 184 227 317)(125 195 228 308)(126 186 229 319)(127 197 230 310)(128 188 231 301)(129 199 232 312)(130 190 233 303)(131 181 234 314)(132 192 235 305)(133 183 236 316)(134 194 237 307)(135 185 238 318)(136 196 239 309)(137 187 240 320)(138 198 221 311)(139 189 222 302)(140 200 223 313)
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), (1,200,175,279,11,190,165,269)(2,189,176,268,12,199,166,278)(3,198,177,277,13,188,167,267)(4,187,178,266,14,197,168,276)(5,196,179,275,15,186,169,265)(6,185,180,264,16,195,170,274)(7,194,161,273,17,184,171,263)(8,183,162,262,18,193,172,272)(9,192,163,271,19,182,173,261)(10,181,164,280,20,191,174,270)(21,288,308,143,31,298,318,153)(22,297,309,152,32,287,319,142)(23,286,310,141,33,296,320,151)(24,295,311,150,34,285,301,160)(25,284,312,159,35,294,302,149)(26,293,313,148,36,283,303,158)(27,282,314,157,37,292,304,147)(28,291,315,146,38,281,305,156)(29,300,316,155,39,290,306,145)(30,289,317,144,40,299,307,154)(41,204,250,239,51,214,260,229)(42,213,251,228,52,203,241,238)(43,202,252,237,53,212,242,227)(44,211,253,226,54,201,243,236)(45,220,254,235,55,210,244,225)(46,209,255,224,56,219,245,234)(47,218,256,233,57,208,246,223)(48,207,257,222,58,217,247,232)(49,216,258,231,59,206,248,221)(50,205,259,240,60,215,249,230)(61,127,112,94,71,137,102,84)(62,136,113,83,72,126,103,93)(63,125,114,92,73,135,104,82)(64,134,115,81,74,124,105,91)(65,123,116,90,75,133,106,100)(66,132,117,99,76,122,107,89)(67,121,118,88,77,131,108,98)(68,130,119,97,78,140,109,87)(69,139,120,86,79,129,110,96)(70,128,101,95,80,138,111,85), (1,251,293,63)(2,242,294,74)(3,253,295,65)(4,244,296,76)(5,255,297,67)(6,246,298,78)(7,257,299,69)(8,248,300,80)(9,259,281,71)(10,250,282,62)(11,241,283,73)(12,252,284,64)(13,243,285,75)(14,254,286,66)(15,245,287,77)(16,256,288,68)(17,247,289,79)(18,258,290,70)(19,249,291,61)(20,260,292,72)(21,92,264,203)(22,83,265,214)(23,94,266,205)(24,85,267,216)(25,96,268,207)(26,87,269,218)(27,98,270,209)(28,89,271,220)(29,100,272,211)(30,91,273,202)(31,82,274,213)(32,93,275,204)(33,84,276,215)(34,95,277,206)(35,86,278,217)(36,97,279,208)(37,88,280,219)(38,99,261,210)(39,90,262,201)(40,81,263,212)(41,157,103,164)(42,148,104,175)(43,159,105,166)(44,150,106,177)(45,141,107,168)(46,152,108,179)(47,143,109,170)(48,154,110,161)(49,145,111,172)(50,156,112,163)(51,147,113,174)(52,158,114,165)(53,149,115,176)(54,160,116,167)(55,151,117,178)(56,142,118,169)(57,153,119,180)(58,144,120,171)(59,155,101,162)(60,146,102,173)(121,191,224,304)(122,182,225,315)(123,193,226,306)(124,184,227,317)(125,195,228,308)(126,186,229,319)(127,197,230,310)(128,188,231,301)(129,199,232,312)(130,190,233,303)(131,181,234,314)(132,192,235,305)(133,183,236,316)(134,194,237,307)(135,185,238,318)(136,196,239,309)(137,187,240,320)(138,198,221,311)(139,189,222,302)(140,200,223,313)>;
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), (1,200,175,279,11,190,165,269)(2,189,176,268,12,199,166,278)(3,198,177,277,13,188,167,267)(4,187,178,266,14,197,168,276)(5,196,179,275,15,186,169,265)(6,185,180,264,16,195,170,274)(7,194,161,273,17,184,171,263)(8,183,162,262,18,193,172,272)(9,192,163,271,19,182,173,261)(10,181,164,280,20,191,174,270)(21,288,308,143,31,298,318,153)(22,297,309,152,32,287,319,142)(23,286,310,141,33,296,320,151)(24,295,311,150,34,285,301,160)(25,284,312,159,35,294,302,149)(26,293,313,148,36,283,303,158)(27,282,314,157,37,292,304,147)(28,291,315,146,38,281,305,156)(29,300,316,155,39,290,306,145)(30,289,317,144,40,299,307,154)(41,204,250,239,51,214,260,229)(42,213,251,228,52,203,241,238)(43,202,252,237,53,212,242,227)(44,211,253,226,54,201,243,236)(45,220,254,235,55,210,244,225)(46,209,255,224,56,219,245,234)(47,218,256,233,57,208,246,223)(48,207,257,222,58,217,247,232)(49,216,258,231,59,206,248,221)(50,205,259,240,60,215,249,230)(61,127,112,94,71,137,102,84)(62,136,113,83,72,126,103,93)(63,125,114,92,73,135,104,82)(64,134,115,81,74,124,105,91)(65,123,116,90,75,133,106,100)(66,132,117,99,76,122,107,89)(67,121,118,88,77,131,108,98)(68,130,119,97,78,140,109,87)(69,139,120,86,79,129,110,96)(70,128,101,95,80,138,111,85), (1,251,293,63)(2,242,294,74)(3,253,295,65)(4,244,296,76)(5,255,297,67)(6,246,298,78)(7,257,299,69)(8,248,300,80)(9,259,281,71)(10,250,282,62)(11,241,283,73)(12,252,284,64)(13,243,285,75)(14,254,286,66)(15,245,287,77)(16,256,288,68)(17,247,289,79)(18,258,290,70)(19,249,291,61)(20,260,292,72)(21,92,264,203)(22,83,265,214)(23,94,266,205)(24,85,267,216)(25,96,268,207)(26,87,269,218)(27,98,270,209)(28,89,271,220)(29,100,272,211)(30,91,273,202)(31,82,274,213)(32,93,275,204)(33,84,276,215)(34,95,277,206)(35,86,278,217)(36,97,279,208)(37,88,280,219)(38,99,261,210)(39,90,262,201)(40,81,263,212)(41,157,103,164)(42,148,104,175)(43,159,105,166)(44,150,106,177)(45,141,107,168)(46,152,108,179)(47,143,109,170)(48,154,110,161)(49,145,111,172)(50,156,112,163)(51,147,113,174)(52,158,114,165)(53,149,115,176)(54,160,116,167)(55,151,117,178)(56,142,118,169)(57,153,119,180)(58,144,120,171)(59,155,101,162)(60,146,102,173)(121,191,224,304)(122,182,225,315)(123,193,226,306)(124,184,227,317)(125,195,228,308)(126,186,229,319)(127,197,230,310)(128,188,231,301)(129,199,232,312)(130,190,233,303)(131,181,234,314)(132,192,235,305)(133,183,236,316)(134,194,237,307)(135,185,238,318)(136,196,239,309)(137,187,240,320)(138,198,221,311)(139,189,222,302)(140,200,223,313) );
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)], [(1,200,175,279,11,190,165,269),(2,189,176,268,12,199,166,278),(3,198,177,277,13,188,167,267),(4,187,178,266,14,197,168,276),(5,196,179,275,15,186,169,265),(6,185,180,264,16,195,170,274),(7,194,161,273,17,184,171,263),(8,183,162,262,18,193,172,272),(9,192,163,271,19,182,173,261),(10,181,164,280,20,191,174,270),(21,288,308,143,31,298,318,153),(22,297,309,152,32,287,319,142),(23,286,310,141,33,296,320,151),(24,295,311,150,34,285,301,160),(25,284,312,159,35,294,302,149),(26,293,313,148,36,283,303,158),(27,282,314,157,37,292,304,147),(28,291,315,146,38,281,305,156),(29,300,316,155,39,290,306,145),(30,289,317,144,40,299,307,154),(41,204,250,239,51,214,260,229),(42,213,251,228,52,203,241,238),(43,202,252,237,53,212,242,227),(44,211,253,226,54,201,243,236),(45,220,254,235,55,210,244,225),(46,209,255,224,56,219,245,234),(47,218,256,233,57,208,246,223),(48,207,257,222,58,217,247,232),(49,216,258,231,59,206,248,221),(50,205,259,240,60,215,249,230),(61,127,112,94,71,137,102,84),(62,136,113,83,72,126,103,93),(63,125,114,92,73,135,104,82),(64,134,115,81,74,124,105,91),(65,123,116,90,75,133,106,100),(66,132,117,99,76,122,107,89),(67,121,118,88,77,131,108,98),(68,130,119,97,78,140,109,87),(69,139,120,86,79,129,110,96),(70,128,101,95,80,138,111,85)], [(1,251,293,63),(2,242,294,74),(3,253,295,65),(4,244,296,76),(5,255,297,67),(6,246,298,78),(7,257,299,69),(8,248,300,80),(9,259,281,71),(10,250,282,62),(11,241,283,73),(12,252,284,64),(13,243,285,75),(14,254,286,66),(15,245,287,77),(16,256,288,68),(17,247,289,79),(18,258,290,70),(19,249,291,61),(20,260,292,72),(21,92,264,203),(22,83,265,214),(23,94,266,205),(24,85,267,216),(25,96,268,207),(26,87,269,218),(27,98,270,209),(28,89,271,220),(29,100,272,211),(30,91,273,202),(31,82,274,213),(32,93,275,204),(33,84,276,215),(34,95,277,206),(35,86,278,217),(36,97,279,208),(37,88,280,219),(38,99,261,210),(39,90,262,201),(40,81,263,212),(41,157,103,164),(42,148,104,175),(43,159,105,166),(44,150,106,177),(45,141,107,168),(46,152,108,179),(47,143,109,170),(48,154,110,161),(49,145,111,172),(50,156,112,163),(51,147,113,174),(52,158,114,165),(53,149,115,176),(54,160,116,167),(55,151,117,178),(56,142,118,169),(57,153,119,180),(58,144,120,171),(59,155,101,162),(60,146,102,173),(121,191,224,304),(122,182,225,315),(123,193,226,306),(124,184,227,317),(125,195,228,308),(126,186,229,319),(127,197,230,310),(128,188,231,301),(129,199,232,312),(130,190,233,303),(131,181,234,314),(132,192,235,305),(133,183,236,316),(134,194,237,307),(135,185,238,318),(136,196,239,309),(137,187,240,320),(138,198,221,311),(139,189,222,302),(140,200,223,313)]])
68 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 5A | 5B | 8A | ··· | 8H | 10A | ··· | 10N | 20A | ··· | 20X |
| order | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 20 | 20 | 20 | 20 | 2 | 2 | 10 | ··· | 10 | 2 | ··· | 2 | 4 | ··· | 4 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | - | + | + | + | - | - | + | - | + | + | - | + | - | |||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | Q8 | D4 | D5 | D8 | SD16 | Q16 | Dic5 | D10 | Dic10 | C4×D5 | D20 | C5⋊D4 | C5⋊D4 | D4⋊D5 | D4.D5 | Q8⋊D5 | C5⋊Q16 |
| kernel | C20.31C42 | C22×C5⋊2C8 | C2×C4⋊Dic5 | C10×C4⋊C4 | C2×C5⋊2C8 | C4⋊Dic5 | C5×C4⋊C4 | C2×C20 | C2×C20 | C22×C10 | C2×C4⋊C4 | C2×C10 | C2×C10 | C2×C10 | C4⋊C4 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C2×C4 | C23 | C22 | C22 | C22 | C22 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 2 | 1 | 1 | 2 | 2 | 4 | 2 | 4 | 2 | 4 | 8 | 4 | 4 | 4 | 2 | 2 | 2 | 2 |
Matrix representation of C20.31C42 ►in GL6(𝔽41)
| 6 | 40 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 33 | 34 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 2 |
| 0 | 0 | 0 | 0 | 40 | 40 |
| 28 | 28 | 0 | 0 | 0 | 0 |
| 32 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 10 | 0 | 0 |
| 0 | 0 | 9 | 27 | 0 | 0 |
| 0 | 0 | 0 | 0 | 30 | 30 |
| 0 | 0 | 0 | 0 | 26 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 13 | 0 | 0 |
| 0 | 0 | 19 | 30 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 11 |
| 0 | 0 | 0 | 0 | 15 | 30 |
G:=sub<GL(6,GF(41))| [6,1,0,0,0,0,40,0,0,0,0,0,0,0,1,33,0,0,0,0,1,34,0,0,0,0,0,0,1,40,0,0,0,0,2,40],[28,32,0,0,0,0,28,13,0,0,0,0,0,0,14,9,0,0,0,0,10,27,0,0,0,0,0,0,30,26,0,0,0,0,30,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,11,19,0,0,0,0,13,30,0,0,0,0,0,0,11,15,0,0,0,0,11,30] >;
C20.31C42 in GAP, Magma, Sage, TeX
C_{20}._{31}C_4^2 % in TeX
G:=Group("C20.31C4^2"); // GroupNames label
G:=SmallGroup(320,87);
// by ID
G=gap.SmallGroup(320,87);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,365,36,570,136,12550]);
// Polycyclic
G:=Group<a,b,c|a^20=c^4=1,b^4=a^10,b*a*b^-1=a^9,c*a*c^-1=a^11,c*b*c^-1=a^15*b>;
// generators/relations