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## G = C42×C20order 320 = 26·5

### Abelian group of type [4,4,20]

Aliases: C42×C20, SmallGroup(320,875)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C42×C20
 Chief series C1 — C2 — C22 — C23 — C22×C10 — C22×C20 — C2×C4×C20 — C42×C20
 Lower central C1 — C42×C20
 Upper central C1 — C42×C20

Generators and relations for C42×C20
G = < a,b,c | a4=b4=c20=1, ab=ba, ac=ca, bc=cb >

Subgroups: 258, all normal (6 characteristic)
C1, C2, C4, C22, C5, C2×C4, C23, C10, C42, C22×C4, C20, C2×C10, C2×C42, C2×C20, C22×C10, C43, C4×C20, C22×C20, C2×C4×C20, C42×C20
Quotients: C1, C2, C4, C22, C5, C2×C4, C23, C10, C42, C22×C4, C20, C2×C10, C2×C42, C2×C20, C22×C10, C43, C4×C20, C22×C20, C2×C4×C20, C42×C20

Smallest permutation representation of C42×C20
Regular action on 320 points
Generators in S320
(1 93 29 286)(2 94 30 287)(3 95 31 288)(4 96 32 289)(5 97 33 290)(6 98 34 291)(7 99 35 292)(8 100 36 293)(9 81 37 294)(10 82 38 295)(11 83 39 296)(12 84 40 297)(13 85 21 298)(14 86 22 299)(15 87 23 300)(16 88 24 281)(17 89 25 282)(18 90 26 283)(19 91 27 284)(20 92 28 285)(41 160 216 72)(42 141 217 73)(43 142 218 74)(44 143 219 75)(45 144 220 76)(46 145 201 77)(47 146 202 78)(48 147 203 79)(49 148 204 80)(50 149 205 61)(51 150 206 62)(52 151 207 63)(53 152 208 64)(54 153 209 65)(55 154 210 66)(56 155 211 67)(57 156 212 68)(58 157 213 69)(59 158 214 70)(60 159 215 71)(101 251 269 137)(102 252 270 138)(103 253 271 139)(104 254 272 140)(105 255 273 121)(106 256 274 122)(107 257 275 123)(108 258 276 124)(109 259 277 125)(110 260 278 126)(111 241 279 127)(112 242 280 128)(113 243 261 129)(114 244 262 130)(115 245 263 131)(116 246 264 132)(117 247 265 133)(118 248 266 134)(119 249 267 135)(120 250 268 136)(161 240 189 304)(162 221 190 305)(163 222 191 306)(164 223 192 307)(165 224 193 308)(166 225 194 309)(167 226 195 310)(168 227 196 311)(169 228 197 312)(170 229 198 313)(171 230 199 314)(172 231 200 315)(173 232 181 316)(174 233 182 317)(175 234 183 318)(176 235 184 319)(177 236 185 320)(178 237 186 301)(179 238 187 302)(180 239 188 303)
(1 158 263 238)(2 159 264 239)(3 160 265 240)(4 141 266 221)(5 142 267 222)(6 143 268 223)(7 144 269 224)(8 145 270 225)(9 146 271 226)(10 147 272 227)(11 148 273 228)(12 149 274 229)(13 150 275 230)(14 151 276 231)(15 152 277 232)(16 153 278 233)(17 154 279 234)(18 155 280 235)(19 156 261 236)(20 157 262 237)(21 62 107 314)(22 63 108 315)(23 64 109 316)(24 65 110 317)(25 66 111 318)(26 67 112 319)(27 68 113 320)(28 69 114 301)(29 70 115 302)(30 71 116 303)(31 72 117 304)(32 73 118 305)(33 74 119 306)(34 75 120 307)(35 76 101 308)(36 77 102 309)(37 78 103 310)(38 79 104 311)(39 80 105 312)(40 61 106 313)(41 247 161 288)(42 248 162 289)(43 249 163 290)(44 250 164 291)(45 251 165 292)(46 252 166 293)(47 253 167 294)(48 254 168 295)(49 255 169 296)(50 256 170 297)(51 257 171 298)(52 258 172 299)(53 259 173 300)(54 260 174 281)(55 241 175 282)(56 242 176 283)(57 243 177 284)(58 244 178 285)(59 245 179 286)(60 246 180 287)(81 202 139 195)(82 203 140 196)(83 204 121 197)(84 205 122 198)(85 206 123 199)(86 207 124 200)(87 208 125 181)(88 209 126 182)(89 210 127 183)(90 211 128 184)(91 212 129 185)(92 213 130 186)(93 214 131 187)(94 215 132 188)(95 216 133 189)(96 217 134 190)(97 218 135 191)(98 219 136 192)(99 220 137 193)(100 201 138 194)
(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)

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

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

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

320 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4BD 5A 5B 5C 5D 10A ··· 10AB 20A ··· 20HP order 1 2 ··· 2 4 ··· 4 5 5 5 5 10 ··· 10 20 ··· 20 size 1 1 ··· 1 1 ··· 1 1 1 1 1 1 ··· 1 1 ··· 1

320 irreducible representations

 dim 1 1 1 1 1 1 type + + image C1 C2 C4 C5 C10 C20 kernel C42×C20 C2×C4×C20 C4×C20 C43 C2×C42 C42 # reps 1 7 56 4 28 224

Matrix representation of C42×C20 in GL3(𝔽41) generated by

 9 0 0 0 40 0 0 0 1
,
 1 0 0 0 9 0 0 0 40
,
 32 0 0 0 39 0 0 0 32
G:=sub<GL(3,GF(41))| [9,0,0,0,40,0,0,0,1],[1,0,0,0,9,0,0,0,40],[32,0,0,0,39,0,0,0,32] >;

C42×C20 in GAP, Magma, Sage, TeX

C_4^2\times C_{20}
% in TeX

G:=Group("C4^2xC20");
// GroupNames label

G:=SmallGroup(320,875);
// by ID

G=gap.SmallGroup(320,875);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,280,568,856]);
// Polycyclic

G:=Group<a,b,c|a^4=b^4=c^20=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations

׿
×
𝔽