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## G = C4⋊C4×C20order 320 = 26·5

### Direct product of C20 and C4⋊C4

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C4⋊C4×C20
 Chief series C1 — C2 — C22 — C23 — C22×C10 — C22×C20 — C5×C2.C42 — C4⋊C4×C20
 Lower central C1 — C2 — C4⋊C4×C20
 Upper central C1 — C22×C20 — C4⋊C4×C20

Generators and relations for C4⋊C4×C20
G = < a,b,c | a20=b4=c4=1, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 242 in 194 conjugacy classes, 146 normal (26 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C23, C10, C10, C42, C42, C4⋊C4, C22×C4, C22×C4, C20, C20, C2×C10, C2×C10, C2.C42, C2×C42, C2×C42, C2×C4⋊C4, C2×C20, C2×C20, C22×C10, C4×C4⋊C4, C4×C20, C4×C20, C5×C4⋊C4, C22×C20, C22×C20, C5×C2.C42, C2×C4×C20, C2×C4×C20, C10×C4⋊C4, C4⋊C4×C20
Quotients:

Smallest permutation representation of C4⋊C4×C20
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)
(1 127 243 70)(2 128 244 71)(3 129 245 72)(4 130 246 73)(5 131 247 74)(6 132 248 75)(7 133 249 76)(8 134 250 77)(9 135 251 78)(10 136 252 79)(11 137 253 80)(12 138 254 61)(13 139 255 62)(14 140 256 63)(15 121 257 64)(16 122 258 65)(17 123 259 66)(18 124 260 67)(19 125 241 68)(20 126 242 69)(21 317 100 215)(22 318 81 216)(23 319 82 217)(24 320 83 218)(25 301 84 219)(26 302 85 220)(27 303 86 201)(28 304 87 202)(29 305 88 203)(30 306 89 204)(31 307 90 205)(32 308 91 206)(33 309 92 207)(34 310 93 208)(35 311 94 209)(36 312 95 210)(37 313 96 211)(38 314 97 212)(39 315 98 213)(40 316 99 214)(41 101 188 235)(42 102 189 236)(43 103 190 237)(44 104 191 238)(45 105 192 239)(46 106 193 240)(47 107 194 221)(48 108 195 222)(49 109 196 223)(50 110 197 224)(51 111 198 225)(52 112 199 226)(53 113 200 227)(54 114 181 228)(55 115 182 229)(56 116 183 230)(57 117 184 231)(58 118 185 232)(59 119 186 233)(60 120 187 234)(141 266 164 293)(142 267 165 294)(143 268 166 295)(144 269 167 296)(145 270 168 297)(146 271 169 298)(147 272 170 299)(148 273 171 300)(149 274 172 281)(150 275 173 282)(151 276 174 283)(152 277 175 284)(153 278 176 285)(154 279 177 286)(155 280 178 287)(156 261 179 288)(157 262 180 289)(158 263 161 290)(159 264 162 291)(160 265 163 292)
(1 49 306 272)(2 50 307 273)(3 51 308 274)(4 52 309 275)(5 53 310 276)(6 54 311 277)(7 55 312 278)(8 56 313 279)(9 57 314 280)(10 58 315 261)(11 59 316 262)(12 60 317 263)(13 41 318 264)(14 42 319 265)(15 43 320 266)(16 44 301 267)(17 45 302 268)(18 46 303 269)(19 47 304 270)(20 48 305 271)(21 161 61 120)(22 162 62 101)(23 163 63 102)(24 164 64 103)(25 165 65 104)(26 166 66 105)(27 167 67 106)(28 168 68 107)(29 169 69 108)(30 170 70 109)(31 171 71 110)(32 172 72 111)(33 173 73 112)(34 174 74 113)(35 175 75 114)(36 176 76 115)(37 177 77 116)(38 178 78 117)(39 179 79 118)(40 180 80 119)(81 159 139 235)(82 160 140 236)(83 141 121 237)(84 142 122 238)(85 143 123 239)(86 144 124 240)(87 145 125 221)(88 146 126 222)(89 147 127 223)(90 148 128 224)(91 149 129 225)(92 150 130 226)(93 151 131 227)(94 152 132 228)(95 153 133 229)(96 154 134 230)(97 155 135 231)(98 156 136 232)(99 157 137 233)(100 158 138 234)(181 209 284 248)(182 210 285 249)(183 211 286 250)(184 212 287 251)(185 213 288 252)(186 214 289 253)(187 215 290 254)(188 216 291 255)(189 217 292 256)(190 218 293 257)(191 219 294 258)(192 220 295 259)(193 201 296 260)(194 202 297 241)(195 203 298 242)(196 204 299 243)(197 205 300 244)(198 206 281 245)(199 207 282 246)(200 208 283 247)

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

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

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

200 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4H 4I ··· 4AF 5A 5B 5C 5D 10A ··· 10AB 20A ··· 20AF 20AG ··· 20DX order 1 2 ··· 2 4 ··· 4 4 ··· 4 5 5 5 5 10 ··· 10 20 ··· 20 20 ··· 20 size 1 1 ··· 1 1 ··· 1 2 ··· 2 1 1 1 1 1 ··· 1 1 ··· 1 2 ··· 2

200 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + - image C1 C2 C2 C2 C4 C4 C5 C10 C10 C10 C20 C20 D4 Q8 C4○D4 C5×D4 C5×Q8 C5×C4○D4 kernel C4⋊C4×C20 C5×C2.C42 C2×C4×C20 C10×C4⋊C4 C4×C20 C5×C4⋊C4 C4×C4⋊C4 C2.C42 C2×C42 C2×C4⋊C4 C42 C4⋊C4 C2×C20 C2×C20 C2×C10 C2×C4 C2×C4 C22 # reps 1 2 3 2 8 16 4 8 12 8 32 64 2 2 4 8 8 16

Matrix representation of C4⋊C4×C20 in GL4(𝔽41) generated by

 32 0 0 0 0 32 0 0 0 0 39 0 0 0 0 39
,
 40 0 0 0 0 40 0 0 0 0 3 25 0 0 16 38
,
 32 0 0 0 0 1 0 0 0 0 0 9 0 0 9 0
G:=sub<GL(4,GF(41))| [32,0,0,0,0,32,0,0,0,0,39,0,0,0,0,39],[40,0,0,0,0,40,0,0,0,0,3,16,0,0,25,38],[32,0,0,0,0,1,0,0,0,0,0,9,0,0,9,0] >;

C4⋊C4×C20 in GAP, Magma, Sage, TeX

C_4\rtimes C_4\times C_{20}
% in TeX

G:=Group("C4:C4xC20");
// GroupNames label

G:=SmallGroup(320,879);
// by ID

G=gap.SmallGroup(320,879);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,560,589,1128,646]);
// Polycyclic

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

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