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G = C42×C20order 320 = 26·5

Abelian group of type [4,4,20]

direct product, abelian, monomial, 2-elementary

Aliases: C42×C20, SmallGroup(320,875)

Series: Derived Chief Lower central Upper central

C1 — C42×C20
C1C2C22C23C22×C10C22×C20C2×C4×C20 — C42×C20
C1 — C42×C20
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 [×7], C4 [×28], C22 [×7], C5, C2×C4 [×42], C23, C10 [×7], C42 [×28], C22×C4 [×7], C20 [×28], C2×C10 [×7], C2×C42 [×7], C2×C20 [×42], C22×C10, C43, C4×C20 [×28], C22×C20 [×7], C2×C4×C20 [×7], C42×C20
Quotients: C1, C2 [×7], C4 [×28], C22 [×7], C5, C2×C4 [×42], C23, C10 [×7], C42 [×28], C22×C4 [×7], C20 [×28], C2×C10 [×7], C2×C42 [×7], C2×C20 [×42], C22×C10, C43, C4×C20 [×28], C22×C20 [×7], C2×C4×C20 [×7], C42×C20

Smallest permutation representation of C42×C20
Regular action on 320 points
Generators in S320
(1 248 49 320)(2 249 50 301)(3 250 51 302)(4 251 52 303)(5 252 53 304)(6 253 54 305)(7 254 55 306)(8 255 56 307)(9 256 57 308)(10 257 58 309)(11 258 59 310)(12 259 60 311)(13 260 41 312)(14 241 42 313)(15 242 43 314)(16 243 44 315)(17 244 45 316)(18 245 46 317)(19 246 47 318)(20 247 48 319)(21 148 97 186)(22 149 98 187)(23 150 99 188)(24 151 100 189)(25 152 81 190)(26 153 82 191)(27 154 83 192)(28 155 84 193)(29 156 85 194)(30 157 86 195)(31 158 87 196)(32 159 88 197)(33 160 89 198)(34 141 90 199)(35 142 91 200)(36 143 92 181)(37 144 93 182)(38 145 94 183)(39 146 95 184)(40 147 96 185)(61 231 280 139)(62 232 261 140)(63 233 262 121)(64 234 263 122)(65 235 264 123)(66 236 265 124)(67 237 266 125)(68 238 267 126)(69 239 268 127)(70 240 269 128)(71 221 270 129)(72 222 271 130)(73 223 272 131)(74 224 273 132)(75 225 274 133)(76 226 275 134)(77 227 276 135)(78 228 277 136)(79 229 278 137)(80 230 279 138)(101 172 202 281)(102 173 203 282)(103 174 204 283)(104 175 205 284)(105 176 206 285)(106 177 207 286)(107 178 208 287)(108 179 209 288)(109 180 210 289)(110 161 211 290)(111 162 212 291)(112 163 213 292)(113 164 214 293)(114 165 215 294)(115 166 216 295)(116 167 217 296)(117 168 218 297)(118 169 219 298)(119 170 220 299)(120 171 201 300)
(1 197 232 178)(2 198 233 179)(3 199 234 180)(4 200 235 161)(5 181 236 162)(6 182 237 163)(7 183 238 164)(8 184 239 165)(9 185 240 166)(10 186 221 167)(11 187 222 168)(12 188 223 169)(13 189 224 170)(14 190 225 171)(15 191 226 172)(16 192 227 173)(17 193 228 174)(18 194 229 175)(19 195 230 176)(20 196 231 177)(21 270 217 257)(22 271 218 258)(23 272 219 259)(24 273 220 260)(25 274 201 241)(26 275 202 242)(27 276 203 243)(28 277 204 244)(29 278 205 245)(30 279 206 246)(31 280 207 247)(32 261 208 248)(33 262 209 249)(34 263 210 250)(35 264 211 251)(36 265 212 252)(37 266 213 253)(38 267 214 254)(39 268 215 255)(40 269 216 256)(41 151 132 299)(42 152 133 300)(43 153 134 281)(44 154 135 282)(45 155 136 283)(46 156 137 284)(47 157 138 285)(48 158 139 286)(49 159 140 287)(50 160 121 288)(51 141 122 289)(52 142 123 290)(53 143 124 291)(54 144 125 292)(55 145 126 293)(56 146 127 294)(57 147 128 295)(58 148 129 296)(59 149 130 297)(60 150 131 298)(61 106 319 87)(62 107 320 88)(63 108 301 89)(64 109 302 90)(65 110 303 91)(66 111 304 92)(67 112 305 93)(68 113 306 94)(69 114 307 95)(70 115 308 96)(71 116 309 97)(72 117 310 98)(73 118 311 99)(74 119 312 100)(75 120 313 81)(76 101 314 82)(77 102 315 83)(78 103 316 84)(79 104 317 85)(80 105 318 86)
(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)| (1,248,49,320)(2,249,50,301)(3,250,51,302)(4,251,52,303)(5,252,53,304)(6,253,54,305)(7,254,55,306)(8,255,56,307)(9,256,57,308)(10,257,58,309)(11,258,59,310)(12,259,60,311)(13,260,41,312)(14,241,42,313)(15,242,43,314)(16,243,44,315)(17,244,45,316)(18,245,46,317)(19,246,47,318)(20,247,48,319)(21,148,97,186)(22,149,98,187)(23,150,99,188)(24,151,100,189)(25,152,81,190)(26,153,82,191)(27,154,83,192)(28,155,84,193)(29,156,85,194)(30,157,86,195)(31,158,87,196)(32,159,88,197)(33,160,89,198)(34,141,90,199)(35,142,91,200)(36,143,92,181)(37,144,93,182)(38,145,94,183)(39,146,95,184)(40,147,96,185)(61,231,280,139)(62,232,261,140)(63,233,262,121)(64,234,263,122)(65,235,264,123)(66,236,265,124)(67,237,266,125)(68,238,267,126)(69,239,268,127)(70,240,269,128)(71,221,270,129)(72,222,271,130)(73,223,272,131)(74,224,273,132)(75,225,274,133)(76,226,275,134)(77,227,276,135)(78,228,277,136)(79,229,278,137)(80,230,279,138)(101,172,202,281)(102,173,203,282)(103,174,204,283)(104,175,205,284)(105,176,206,285)(106,177,207,286)(107,178,208,287)(108,179,209,288)(109,180,210,289)(110,161,211,290)(111,162,212,291)(112,163,213,292)(113,164,214,293)(114,165,215,294)(115,166,216,295)(116,167,217,296)(117,168,218,297)(118,169,219,298)(119,170,220,299)(120,171,201,300), (1,197,232,178)(2,198,233,179)(3,199,234,180)(4,200,235,161)(5,181,236,162)(6,182,237,163)(7,183,238,164)(8,184,239,165)(9,185,240,166)(10,186,221,167)(11,187,222,168)(12,188,223,169)(13,189,224,170)(14,190,225,171)(15,191,226,172)(16,192,227,173)(17,193,228,174)(18,194,229,175)(19,195,230,176)(20,196,231,177)(21,270,217,257)(22,271,218,258)(23,272,219,259)(24,273,220,260)(25,274,201,241)(26,275,202,242)(27,276,203,243)(28,277,204,244)(29,278,205,245)(30,279,206,246)(31,280,207,247)(32,261,208,248)(33,262,209,249)(34,263,210,250)(35,264,211,251)(36,265,212,252)(37,266,213,253)(38,267,214,254)(39,268,215,255)(40,269,216,256)(41,151,132,299)(42,152,133,300)(43,153,134,281)(44,154,135,282)(45,155,136,283)(46,156,137,284)(47,157,138,285)(48,158,139,286)(49,159,140,287)(50,160,121,288)(51,141,122,289)(52,142,123,290)(53,143,124,291)(54,144,125,292)(55,145,126,293)(56,146,127,294)(57,147,128,295)(58,148,129,296)(59,149,130,297)(60,150,131,298)(61,106,319,87)(62,107,320,88)(63,108,301,89)(64,109,302,90)(65,110,303,91)(66,111,304,92)(67,112,305,93)(68,113,306,94)(69,114,307,95)(70,115,308,96)(71,116,309,97)(72,117,310,98)(73,118,311,99)(74,119,312,100)(75,120,313,81)(76,101,314,82)(77,102,315,83)(78,103,316,84)(79,104,317,85)(80,105,318,86), (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( (1,248,49,320)(2,249,50,301)(3,250,51,302)(4,251,52,303)(5,252,53,304)(6,253,54,305)(7,254,55,306)(8,255,56,307)(9,256,57,308)(10,257,58,309)(11,258,59,310)(12,259,60,311)(13,260,41,312)(14,241,42,313)(15,242,43,314)(16,243,44,315)(17,244,45,316)(18,245,46,317)(19,246,47,318)(20,247,48,319)(21,148,97,186)(22,149,98,187)(23,150,99,188)(24,151,100,189)(25,152,81,190)(26,153,82,191)(27,154,83,192)(28,155,84,193)(29,156,85,194)(30,157,86,195)(31,158,87,196)(32,159,88,197)(33,160,89,198)(34,141,90,199)(35,142,91,200)(36,143,92,181)(37,144,93,182)(38,145,94,183)(39,146,95,184)(40,147,96,185)(61,231,280,139)(62,232,261,140)(63,233,262,121)(64,234,263,122)(65,235,264,123)(66,236,265,124)(67,237,266,125)(68,238,267,126)(69,239,268,127)(70,240,269,128)(71,221,270,129)(72,222,271,130)(73,223,272,131)(74,224,273,132)(75,225,274,133)(76,226,275,134)(77,227,276,135)(78,228,277,136)(79,229,278,137)(80,230,279,138)(101,172,202,281)(102,173,203,282)(103,174,204,283)(104,175,205,284)(105,176,206,285)(106,177,207,286)(107,178,208,287)(108,179,209,288)(109,180,210,289)(110,161,211,290)(111,162,212,291)(112,163,213,292)(113,164,214,293)(114,165,215,294)(115,166,216,295)(116,167,217,296)(117,168,218,297)(118,169,219,298)(119,170,220,299)(120,171,201,300), (1,197,232,178)(2,198,233,179)(3,199,234,180)(4,200,235,161)(5,181,236,162)(6,182,237,163)(7,183,238,164)(8,184,239,165)(9,185,240,166)(10,186,221,167)(11,187,222,168)(12,188,223,169)(13,189,224,170)(14,190,225,171)(15,191,226,172)(16,192,227,173)(17,193,228,174)(18,194,229,175)(19,195,230,176)(20,196,231,177)(21,270,217,257)(22,271,218,258)(23,272,219,259)(24,273,220,260)(25,274,201,241)(26,275,202,242)(27,276,203,243)(28,277,204,244)(29,278,205,245)(30,279,206,246)(31,280,207,247)(32,261,208,248)(33,262,209,249)(34,263,210,250)(35,264,211,251)(36,265,212,252)(37,266,213,253)(38,267,214,254)(39,268,215,255)(40,269,216,256)(41,151,132,299)(42,152,133,300)(43,153,134,281)(44,154,135,282)(45,155,136,283)(46,156,137,284)(47,157,138,285)(48,158,139,286)(49,159,140,287)(50,160,121,288)(51,141,122,289)(52,142,123,290)(53,143,124,291)(54,144,125,292)(55,145,126,293)(56,146,127,294)(57,147,128,295)(58,148,129,296)(59,149,130,297)(60,150,131,298)(61,106,319,87)(62,107,320,88)(63,108,301,89)(64,109,302,90)(65,110,303,91)(66,111,304,92)(67,112,305,93)(68,113,306,94)(69,114,307,95)(70,115,308,96)(71,116,309,97)(72,117,310,98)(73,118,311,99)(74,119,312,100)(75,120,313,81)(76,101,314,82)(77,102,315,83)(78,103,316,84)(79,104,317,85)(80,105,318,86), (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([(1,248,49,320),(2,249,50,301),(3,250,51,302),(4,251,52,303),(5,252,53,304),(6,253,54,305),(7,254,55,306),(8,255,56,307),(9,256,57,308),(10,257,58,309),(11,258,59,310),(12,259,60,311),(13,260,41,312),(14,241,42,313),(15,242,43,314),(16,243,44,315),(17,244,45,316),(18,245,46,317),(19,246,47,318),(20,247,48,319),(21,148,97,186),(22,149,98,187),(23,150,99,188),(24,151,100,189),(25,152,81,190),(26,153,82,191),(27,154,83,192),(28,155,84,193),(29,156,85,194),(30,157,86,195),(31,158,87,196),(32,159,88,197),(33,160,89,198),(34,141,90,199),(35,142,91,200),(36,143,92,181),(37,144,93,182),(38,145,94,183),(39,146,95,184),(40,147,96,185),(61,231,280,139),(62,232,261,140),(63,233,262,121),(64,234,263,122),(65,235,264,123),(66,236,265,124),(67,237,266,125),(68,238,267,126),(69,239,268,127),(70,240,269,128),(71,221,270,129),(72,222,271,130),(73,223,272,131),(74,224,273,132),(75,225,274,133),(76,226,275,134),(77,227,276,135),(78,228,277,136),(79,229,278,137),(80,230,279,138),(101,172,202,281),(102,173,203,282),(103,174,204,283),(104,175,205,284),(105,176,206,285),(106,177,207,286),(107,178,208,287),(108,179,209,288),(109,180,210,289),(110,161,211,290),(111,162,212,291),(112,163,213,292),(113,164,214,293),(114,165,215,294),(115,166,216,295),(116,167,217,296),(117,168,218,297),(118,169,219,298),(119,170,220,299),(120,171,201,300)], [(1,197,232,178),(2,198,233,179),(3,199,234,180),(4,200,235,161),(5,181,236,162),(6,182,237,163),(7,183,238,164),(8,184,239,165),(9,185,240,166),(10,186,221,167),(11,187,222,168),(12,188,223,169),(13,189,224,170),(14,190,225,171),(15,191,226,172),(16,192,227,173),(17,193,228,174),(18,194,229,175),(19,195,230,176),(20,196,231,177),(21,270,217,257),(22,271,218,258),(23,272,219,259),(24,273,220,260),(25,274,201,241),(26,275,202,242),(27,276,203,243),(28,277,204,244),(29,278,205,245),(30,279,206,246),(31,280,207,247),(32,261,208,248),(33,262,209,249),(34,263,210,250),(35,264,211,251),(36,265,212,252),(37,266,213,253),(38,267,214,254),(39,268,215,255),(40,269,216,256),(41,151,132,299),(42,152,133,300),(43,153,134,281),(44,154,135,282),(45,155,136,283),(46,156,137,284),(47,157,138,285),(48,158,139,286),(49,159,140,287),(50,160,121,288),(51,141,122,289),(52,142,123,290),(53,143,124,291),(54,144,125,292),(55,145,126,293),(56,146,127,294),(57,147,128,295),(58,148,129,296),(59,149,130,297),(60,150,131,298),(61,106,319,87),(62,107,320,88),(63,108,301,89),(64,109,302,90),(65,110,303,91),(66,111,304,92),(67,112,305,93),(68,113,306,94),(69,114,307,95),(70,115,308,96),(71,116,309,97),(72,117,310,98),(73,118,311,99),(74,119,312,100),(75,120,313,81),(76,101,314,82),(77,102,315,83),(78,103,316,84),(79,104,317,85),(80,105,318,86)], [(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···2G4A···4BD5A5B5C5D10A···10AB20A···20HP
order12···24···4555510···1020···20
size11···11···111111···11···1

320 irreducible representations

dim111111
type++
imageC1C2C4C5C10C20
kernelC42×C20C2×C4×C20C4×C20C43C2×C42C42
# reps1756428224

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

900
0400
001
,
100
090
0040
,
3200
0390
0032
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