metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20⋊6(C4⋊C4), C4⋊C4⋊6Dic5, C4⋊1(C4⋊Dic5), C2.5(C20⋊Q8), (C2×C20).20Q8, C10.34(C4×Q8), C2.7(D4×Dic5), C2.4(Q8×Dic5), (C2×C20).140D4, C10.119(C4×D4), (C2×C4).142D20, C10.23(C4⋊Q8), C22.25(Q8×D5), C2.3(C4⋊D20), (C2×Dic5).25Q8, (C2×C4).31Dic10, C22.109(D4×D5), C22.46(C2×D20), C10.52(C4⋊D4), C2.4(D10⋊2Q8), (C2×Dic5).154D4, (C22×C4).102D10, C10.48(C22⋊Q8), C2.5(C4.Dic10), (C22×C20).66C22, C10.24(C42.C2), C22.29(C2×Dic10), C23.294(C22×D5), C22.58(D4⋊2D5), (C22×C10).349C23, C5⋊6(C23.65C23), C22.26(Q8⋊2D5), C22.42(C22×Dic5), C10.10C42.29C2, (C22×Dic5).214C22, (C5×C4⋊C4)⋊16C4, C10.61(C2×C4⋊C4), (C2×C4⋊C4).22D5, (C10×C4⋊C4).15C2, C2.8(C2×C4⋊Dic5), (C2×C4×Dic5).8C2, (C2×C10).38(C2×Q8), (C2×C20).218(C2×C4), (C2×C10).333(C2×D4), (C2×C4⋊Dic5).36C2, (C2×C4).18(C2×Dic5), (C2×C10).187(C4○D4), (C2×C10).282(C22×C4), SmallGroup(320,612)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C20⋊6(C4⋊C4)
G = < a,b,c | a20=b4=c4=1, bab-1=a11, cac-1=a9, cbc-1=b-1 >
Subgroups: 510 in 170 conjugacy classes, 91 normal (41 characteristic)
C1, C2, C4, C4, C22, C5, C2×C4, C2×C4, C23, C10, C42, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, Dic5, C20, C20, C2×C10, C2.C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×C10, C23.65C23, C4×Dic5, C4⋊Dic5, C5×C4⋊C4, C22×Dic5, C22×Dic5, C22×C20, C22×C20, C10.10C42, C2×C4×Dic5, C2×C4⋊Dic5, C2×C4⋊Dic5, C10×C4⋊C4, C20⋊6(C4⋊C4)
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D5, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, Dic5, D10, C2×C4⋊C4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C42.C2, C4⋊Q8, Dic10, D20, C2×Dic5, C22×D5, C23.65C23, C4⋊Dic5, C2×Dic10, C2×D20, D4×D5, D4⋊2D5, Q8×D5, Q8⋊2D5, C22×Dic5, C20⋊Q8, C4.Dic10, C4⋊D20, D10⋊2Q8, C2×C4⋊Dic5, D4×Dic5, Q8×Dic5, C20⋊6(C4⋊C4)
►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)
(1 297 70 54)(2 288 71 45)(3 299 72 56)(4 290 73 47)(5 281 74 58)(6 292 75 49)(7 283 76 60)(8 294 77 51)(9 285 78 42)(10 296 79 53)(11 287 80 44)(12 298 61 55)(13 289 62 46)(14 300 63 57)(15 291 64 48)(16 282 65 59)(17 293 66 50)(18 284 67 41)(19 295 68 52)(20 286 69 43)(21 143 315 227)(22 154 316 238)(23 145 317 229)(24 156 318 240)(25 147 319 231)(26 158 320 222)(27 149 301 233)(28 160 302 224)(29 151 303 235)(30 142 304 226)(31 153 305 237)(32 144 306 228)(33 155 307 239)(34 146 308 230)(35 157 309 221)(36 148 310 232)(37 159 311 223)(38 150 312 234)(39 141 313 225)(40 152 314 236)(81 166 107 248)(82 177 108 259)(83 168 109 250)(84 179 110 241)(85 170 111 252)(86 161 112 243)(87 172 113 254)(88 163 114 245)(89 174 115 256)(90 165 116 247)(91 176 117 258)(92 167 118 249)(93 178 119 260)(94 169 120 251)(95 180 101 242)(96 171 102 253)(97 162 103 244)(98 173 104 255)(99 164 105 246)(100 175 106 257)(121 278 209 186)(122 269 210 197)(123 280 211 188)(124 271 212 199)(125 262 213 190)(126 273 214 181)(127 264 215 192)(128 275 216 183)(129 266 217 194)(130 277 218 185)(131 268 219 196)(132 279 220 187)(133 270 201 198)(134 261 202 189)(135 272 203 200)(136 263 204 191)(137 274 205 182)(138 265 206 193)(139 276 207 184)(140 267 208 195)
(1 120 160 211)(2 109 141 220)(3 118 142 209)(4 107 143 218)(5 116 144 207)(6 105 145 216)(7 114 146 205)(8 103 147 214)(9 112 148 203)(10 101 149 212)(11 110 150 201)(12 119 151 210)(13 108 152 219)(14 117 153 208)(15 106 154 217)(16 115 155 206)(17 104 156 215)(18 113 157 204)(19 102 158 213)(20 111 159 202)(21 185 47 248)(22 194 48 257)(23 183 49 246)(24 192 50 255)(25 181 51 244)(26 190 52 253)(27 199 53 242)(28 188 54 251)(29 197 55 260)(30 186 56 249)(31 195 57 258)(32 184 58 247)(33 193 59 256)(34 182 60 245)(35 191 41 254)(36 200 42 243)(37 189 43 252)(38 198 44 241)(39 187 45 250)(40 196 46 259)(61 93 235 122)(62 82 236 131)(63 91 237 140)(64 100 238 129)(65 89 239 138)(66 98 240 127)(67 87 221 136)(68 96 222 125)(69 85 223 134)(70 94 224 123)(71 83 225 132)(72 92 226 121)(73 81 227 130)(74 90 228 139)(75 99 229 128)(76 88 230 137)(77 97 231 126)(78 86 232 135)(79 95 233 124)(80 84 234 133)(161 310 272 285)(162 319 273 294)(163 308 274 283)(164 317 275 292)(165 306 276 281)(166 315 277 290)(167 304 278 299)(168 313 279 288)(169 302 280 297)(170 311 261 286)(171 320 262 295)(172 309 263 284)(173 318 264 293)(174 307 265 282)(175 316 266 291)(176 305 267 300)(177 314 268 289)(178 303 269 298)(179 312 270 287)(180 301 271 296)
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), (1,297,70,54)(2,288,71,45)(3,299,72,56)(4,290,73,47)(5,281,74,58)(6,292,75,49)(7,283,76,60)(8,294,77,51)(9,285,78,42)(10,296,79,53)(11,287,80,44)(12,298,61,55)(13,289,62,46)(14,300,63,57)(15,291,64,48)(16,282,65,59)(17,293,66,50)(18,284,67,41)(19,295,68,52)(20,286,69,43)(21,143,315,227)(22,154,316,238)(23,145,317,229)(24,156,318,240)(25,147,319,231)(26,158,320,222)(27,149,301,233)(28,160,302,224)(29,151,303,235)(30,142,304,226)(31,153,305,237)(32,144,306,228)(33,155,307,239)(34,146,308,230)(35,157,309,221)(36,148,310,232)(37,159,311,223)(38,150,312,234)(39,141,313,225)(40,152,314,236)(81,166,107,248)(82,177,108,259)(83,168,109,250)(84,179,110,241)(85,170,111,252)(86,161,112,243)(87,172,113,254)(88,163,114,245)(89,174,115,256)(90,165,116,247)(91,176,117,258)(92,167,118,249)(93,178,119,260)(94,169,120,251)(95,180,101,242)(96,171,102,253)(97,162,103,244)(98,173,104,255)(99,164,105,246)(100,175,106,257)(121,278,209,186)(122,269,210,197)(123,280,211,188)(124,271,212,199)(125,262,213,190)(126,273,214,181)(127,264,215,192)(128,275,216,183)(129,266,217,194)(130,277,218,185)(131,268,219,196)(132,279,220,187)(133,270,201,198)(134,261,202,189)(135,272,203,200)(136,263,204,191)(137,274,205,182)(138,265,206,193)(139,276,207,184)(140,267,208,195), (1,120,160,211)(2,109,141,220)(3,118,142,209)(4,107,143,218)(5,116,144,207)(6,105,145,216)(7,114,146,205)(8,103,147,214)(9,112,148,203)(10,101,149,212)(11,110,150,201)(12,119,151,210)(13,108,152,219)(14,117,153,208)(15,106,154,217)(16,115,155,206)(17,104,156,215)(18,113,157,204)(19,102,158,213)(20,111,159,202)(21,185,47,248)(22,194,48,257)(23,183,49,246)(24,192,50,255)(25,181,51,244)(26,190,52,253)(27,199,53,242)(28,188,54,251)(29,197,55,260)(30,186,56,249)(31,195,57,258)(32,184,58,247)(33,193,59,256)(34,182,60,245)(35,191,41,254)(36,200,42,243)(37,189,43,252)(38,198,44,241)(39,187,45,250)(40,196,46,259)(61,93,235,122)(62,82,236,131)(63,91,237,140)(64,100,238,129)(65,89,239,138)(66,98,240,127)(67,87,221,136)(68,96,222,125)(69,85,223,134)(70,94,224,123)(71,83,225,132)(72,92,226,121)(73,81,227,130)(74,90,228,139)(75,99,229,128)(76,88,230,137)(77,97,231,126)(78,86,232,135)(79,95,233,124)(80,84,234,133)(161,310,272,285)(162,319,273,294)(163,308,274,283)(164,317,275,292)(165,306,276,281)(166,315,277,290)(167,304,278,299)(168,313,279,288)(169,302,280,297)(170,311,261,286)(171,320,262,295)(172,309,263,284)(173,318,264,293)(174,307,265,282)(175,316,266,291)(176,305,267,300)(177,314,268,289)(178,303,269,298)(179,312,270,287)(180,301,271,296)>;
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), (1,297,70,54)(2,288,71,45)(3,299,72,56)(4,290,73,47)(5,281,74,58)(6,292,75,49)(7,283,76,60)(8,294,77,51)(9,285,78,42)(10,296,79,53)(11,287,80,44)(12,298,61,55)(13,289,62,46)(14,300,63,57)(15,291,64,48)(16,282,65,59)(17,293,66,50)(18,284,67,41)(19,295,68,52)(20,286,69,43)(21,143,315,227)(22,154,316,238)(23,145,317,229)(24,156,318,240)(25,147,319,231)(26,158,320,222)(27,149,301,233)(28,160,302,224)(29,151,303,235)(30,142,304,226)(31,153,305,237)(32,144,306,228)(33,155,307,239)(34,146,308,230)(35,157,309,221)(36,148,310,232)(37,159,311,223)(38,150,312,234)(39,141,313,225)(40,152,314,236)(81,166,107,248)(82,177,108,259)(83,168,109,250)(84,179,110,241)(85,170,111,252)(86,161,112,243)(87,172,113,254)(88,163,114,245)(89,174,115,256)(90,165,116,247)(91,176,117,258)(92,167,118,249)(93,178,119,260)(94,169,120,251)(95,180,101,242)(96,171,102,253)(97,162,103,244)(98,173,104,255)(99,164,105,246)(100,175,106,257)(121,278,209,186)(122,269,210,197)(123,280,211,188)(124,271,212,199)(125,262,213,190)(126,273,214,181)(127,264,215,192)(128,275,216,183)(129,266,217,194)(130,277,218,185)(131,268,219,196)(132,279,220,187)(133,270,201,198)(134,261,202,189)(135,272,203,200)(136,263,204,191)(137,274,205,182)(138,265,206,193)(139,276,207,184)(140,267,208,195), (1,120,160,211)(2,109,141,220)(3,118,142,209)(4,107,143,218)(5,116,144,207)(6,105,145,216)(7,114,146,205)(8,103,147,214)(9,112,148,203)(10,101,149,212)(11,110,150,201)(12,119,151,210)(13,108,152,219)(14,117,153,208)(15,106,154,217)(16,115,155,206)(17,104,156,215)(18,113,157,204)(19,102,158,213)(20,111,159,202)(21,185,47,248)(22,194,48,257)(23,183,49,246)(24,192,50,255)(25,181,51,244)(26,190,52,253)(27,199,53,242)(28,188,54,251)(29,197,55,260)(30,186,56,249)(31,195,57,258)(32,184,58,247)(33,193,59,256)(34,182,60,245)(35,191,41,254)(36,200,42,243)(37,189,43,252)(38,198,44,241)(39,187,45,250)(40,196,46,259)(61,93,235,122)(62,82,236,131)(63,91,237,140)(64,100,238,129)(65,89,239,138)(66,98,240,127)(67,87,221,136)(68,96,222,125)(69,85,223,134)(70,94,224,123)(71,83,225,132)(72,92,226,121)(73,81,227,130)(74,90,228,139)(75,99,229,128)(76,88,230,137)(77,97,231,126)(78,86,232,135)(79,95,233,124)(80,84,234,133)(161,310,272,285)(162,319,273,294)(163,308,274,283)(164,317,275,292)(165,306,276,281)(166,315,277,290)(167,304,278,299)(168,313,279,288)(169,302,280,297)(170,311,261,286)(171,320,262,295)(172,309,263,284)(173,318,264,293)(174,307,265,282)(175,316,266,291)(176,305,267,300)(177,314,268,289)(178,303,269,298)(179,312,270,287)(180,301,271,296) );
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)], [(1,297,70,54),(2,288,71,45),(3,299,72,56),(4,290,73,47),(5,281,74,58),(6,292,75,49),(7,283,76,60),(8,294,77,51),(9,285,78,42),(10,296,79,53),(11,287,80,44),(12,298,61,55),(13,289,62,46),(14,300,63,57),(15,291,64,48),(16,282,65,59),(17,293,66,50),(18,284,67,41),(19,295,68,52),(20,286,69,43),(21,143,315,227),(22,154,316,238),(23,145,317,229),(24,156,318,240),(25,147,319,231),(26,158,320,222),(27,149,301,233),(28,160,302,224),(29,151,303,235),(30,142,304,226),(31,153,305,237),(32,144,306,228),(33,155,307,239),(34,146,308,230),(35,157,309,221),(36,148,310,232),(37,159,311,223),(38,150,312,234),(39,141,313,225),(40,152,314,236),(81,166,107,248),(82,177,108,259),(83,168,109,250),(84,179,110,241),(85,170,111,252),(86,161,112,243),(87,172,113,254),(88,163,114,245),(89,174,115,256),(90,165,116,247),(91,176,117,258),(92,167,118,249),(93,178,119,260),(94,169,120,251),(95,180,101,242),(96,171,102,253),(97,162,103,244),(98,173,104,255),(99,164,105,246),(100,175,106,257),(121,278,209,186),(122,269,210,197),(123,280,211,188),(124,271,212,199),(125,262,213,190),(126,273,214,181),(127,264,215,192),(128,275,216,183),(129,266,217,194),(130,277,218,185),(131,268,219,196),(132,279,220,187),(133,270,201,198),(134,261,202,189),(135,272,203,200),(136,263,204,191),(137,274,205,182),(138,265,206,193),(139,276,207,184),(140,267,208,195)], [(1,120,160,211),(2,109,141,220),(3,118,142,209),(4,107,143,218),(5,116,144,207),(6,105,145,216),(7,114,146,205),(8,103,147,214),(9,112,148,203),(10,101,149,212),(11,110,150,201),(12,119,151,210),(13,108,152,219),(14,117,153,208),(15,106,154,217),(16,115,155,206),(17,104,156,215),(18,113,157,204),(19,102,158,213),(20,111,159,202),(21,185,47,248),(22,194,48,257),(23,183,49,246),(24,192,50,255),(25,181,51,244),(26,190,52,253),(27,199,53,242),(28,188,54,251),(29,197,55,260),(30,186,56,249),(31,195,57,258),(32,184,58,247),(33,193,59,256),(34,182,60,245),(35,191,41,254),(36,200,42,243),(37,189,43,252),(38,198,44,241),(39,187,45,250),(40,196,46,259),(61,93,235,122),(62,82,236,131),(63,91,237,140),(64,100,238,129),(65,89,239,138),(66,98,240,127),(67,87,221,136),(68,96,222,125),(69,85,223,134),(70,94,224,123),(71,83,225,132),(72,92,226,121),(73,81,227,130),(74,90,228,139),(75,99,229,128),(76,88,230,137),(77,97,231,126),(78,86,232,135),(79,95,233,124),(80,84,234,133),(161,310,272,285),(162,319,273,294),(163,308,274,283),(164,317,275,292),(165,306,276,281),(166,315,277,290),(167,304,278,299),(168,313,279,288),(169,302,280,297),(170,311,261,286),(171,320,262,295),(172,309,263,284),(173,318,264,293),(174,307,265,282),(175,316,266,291),(176,305,267,300),(177,314,268,289),(178,303,269,298),(179,312,270,287),(180,301,271,296)]])
68 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4P | 4Q | 4R | 4S | 4T | 5A | 5B | 10A | ··· | 10N | 20A | ··· | 20X |
| order | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | - | + | - | + | - | + | + | - | - | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C4 | D4 | Q8 | D4 | Q8 | D5 | C4○D4 | Dic5 | D10 | Dic10 | D20 | D4×D5 | D4⋊2D5 | Q8×D5 | Q8⋊2D5 |
| kernel | C20⋊6(C4⋊C4) | C10.10C42 | C2×C4×Dic5 | C2×C4⋊Dic5 | C10×C4⋊C4 | C5×C4⋊C4 | C2×Dic5 | C2×Dic5 | C2×C20 | C2×C20 | C2×C4⋊C4 | C2×C10 | C4⋊C4 | C22×C4 | C2×C4 | C2×C4 | C22 | C22 | C22 | C22 |
| # reps | 1 | 2 | 1 | 3 | 1 | 8 | 2 | 2 | 2 | 2 | 2 | 4 | 8 | 6 | 8 | 8 | 2 | 2 | 2 | 2 |
Matrix representation of C20⋊6(C4⋊C4) ►in GL6(𝔽41)
| 26 | 15 | 0 | 0 | 0 | 0 |
| 15 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 1 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 40 | 39 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 30 | 32 |
| 0 | 0 | 0 | 0 | 9 | 11 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 1 | 0 | 0 |
| 0 | 0 | 38 | 24 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 19 | 32 |
G:=sub<GL(6,GF(41))| [26,15,0,0,0,0,15,15,0,0,0,0,0,0,0,40,0,0,0,0,1,7,0,0,0,0,0,0,7,40,0,0,0,0,1,0],[40,0,0,0,0,0,39,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,30,9,0,0,0,0,32,11],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,17,38,0,0,0,0,1,24,0,0,0,0,0,0,9,19,0,0,0,0,0,32] >;
C20⋊6(C4⋊C4) in GAP, Magma, Sage, TeX
C_{20}\rtimes_6(C_4\rtimes C_4) % in TeX
G:=Group("C20:6(C4:C4)"); // GroupNames label
G:=SmallGroup(320,612);
// by ID
G=gap.SmallGroup(320,612);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,232,422,387,100,12550]);
// Polycyclic
G:=Group<a,b,c|a^20=b^4=c^4=1,b*a*b^-1=a^11,c*a*c^-1=a^9,c*b*c^-1=b^-1>;
// generators/relations