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## G = C4×Dic20order 320 = 26·5

### Direct product of C4 and Dic20

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C4×Dic20
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — C2×Dic10 — C2×Dic20 — C4×Dic20
 Lower central C5 — C10 — C20 — C4×Dic20
 Upper central C1 — C2×C4 — C42 — C4×C8

Generators and relations for C4×Dic20
G = < a,b,c | a4=b40=1, c2=b20, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 374 in 110 conjugacy classes, 55 normal (31 characteristic)
C1, C2, C4, C4, C4, C22, C5, C8, C8, C2×C4, C2×C4, Q8, C10, C42, C42, C4⋊C4, C2×C8, Q16, C2×Q8, Dic5, C20, C20, C20, C2×C10, C4×C8, Q8⋊C4, C2.D8, C4×Q8, C2×Q16, C40, C40, Dic10, Dic10, C2×Dic5, C2×C20, C4×Q16, Dic20, C4×Dic5, C10.D4, C4⋊Dic5, C4×C20, C2×C40, C2×Dic10, C20.44D4, C405C4, C4×C40, C4×Dic10, C2×Dic20, C4×Dic20
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, Q16, C22×C4, C2×D4, C4○D4, D10, C4×D4, C2×Q16, C4○D8, C4×D5, D20, C22×D5, C4×Q16, Dic20, C2×C4×D5, C2×D20, C4○D20, C4×D20, D407C2, C2×Dic20, C4×Dic20

Smallest permutation representation of C4×Dic20
Regular action on 320 points
Generators in S320
(1 202 107 46)(2 203 108 47)(3 204 109 48)(4 205 110 49)(5 206 111 50)(6 207 112 51)(7 208 113 52)(8 209 114 53)(9 210 115 54)(10 211 116 55)(11 212 117 56)(12 213 118 57)(13 214 119 58)(14 215 120 59)(15 216 81 60)(16 217 82 61)(17 218 83 62)(18 219 84 63)(19 220 85 64)(20 221 86 65)(21 222 87 66)(22 223 88 67)(23 224 89 68)(24 225 90 69)(25 226 91 70)(26 227 92 71)(27 228 93 72)(28 229 94 73)(29 230 95 74)(30 231 96 75)(31 232 97 76)(32 233 98 77)(33 234 99 78)(34 235 100 79)(35 236 101 80)(36 237 102 41)(37 238 103 42)(38 239 104 43)(39 240 105 44)(40 201 106 45)(121 306 178 247)(122 307 179 248)(123 308 180 249)(124 309 181 250)(125 310 182 251)(126 311 183 252)(127 312 184 253)(128 313 185 254)(129 314 186 255)(130 315 187 256)(131 316 188 257)(132 317 189 258)(133 318 190 259)(134 319 191 260)(135 320 192 261)(136 281 193 262)(137 282 194 263)(138 283 195 264)(139 284 196 265)(140 285 197 266)(141 286 198 267)(142 287 199 268)(143 288 200 269)(144 289 161 270)(145 290 162 271)(146 291 163 272)(147 292 164 273)(148 293 165 274)(149 294 166 275)(150 295 167 276)(151 296 168 277)(152 297 169 278)(153 298 170 279)(154 299 171 280)(155 300 172 241)(156 301 173 242)(157 302 174 243)(158 303 175 244)(159 304 176 245)(160 305 177 246)
(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)
(1 261 21 241)(2 260 22 280)(3 259 23 279)(4 258 24 278)(5 257 25 277)(6 256 26 276)(7 255 27 275)(8 254 28 274)(9 253 29 273)(10 252 30 272)(11 251 31 271)(12 250 32 270)(13 249 33 269)(14 248 34 268)(15 247 35 267)(16 246 36 266)(17 245 37 265)(18 244 38 264)(19 243 39 263)(20 242 40 262)(41 197 61 177)(42 196 62 176)(43 195 63 175)(44 194 64 174)(45 193 65 173)(46 192 66 172)(47 191 67 171)(48 190 68 170)(49 189 69 169)(50 188 70 168)(51 187 71 167)(52 186 72 166)(53 185 73 165)(54 184 74 164)(55 183 75 163)(56 182 76 162)(57 181 77 161)(58 180 78 200)(59 179 79 199)(60 178 80 198)(81 306 101 286)(82 305 102 285)(83 304 103 284)(84 303 104 283)(85 302 105 282)(86 301 106 281)(87 300 107 320)(88 299 108 319)(89 298 109 318)(90 297 110 317)(91 296 111 316)(92 295 112 315)(93 294 113 314)(94 293 114 313)(95 292 115 312)(96 291 116 311)(97 290 117 310)(98 289 118 309)(99 288 119 308)(100 287 120 307)(121 236 141 216)(122 235 142 215)(123 234 143 214)(124 233 144 213)(125 232 145 212)(126 231 146 211)(127 230 147 210)(128 229 148 209)(129 228 149 208)(130 227 150 207)(131 226 151 206)(132 225 152 205)(133 224 153 204)(134 223 154 203)(135 222 155 202)(136 221 156 201)(137 220 157 240)(138 219 158 239)(139 218 159 238)(140 217 160 237)

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

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

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

92 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 4G 4H 4I ··· 4P 5A 5B 8A ··· 8H 10A ··· 10F 20A ··· 20X 40A ··· 40AF order 1 2 2 2 4 4 4 4 4 4 4 4 4 ··· 4 5 5 8 ··· 8 10 ··· 10 20 ··· 20 40 ··· 40 size 1 1 1 1 1 1 1 1 2 2 2 2 20 ··· 20 2 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

92 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + - + + + - image C1 C2 C2 C2 C2 C2 C4 D4 D5 Q16 C4○D4 D10 D10 C4○D8 C4×D5 D20 Dic20 C4○D20 D40⋊7C2 kernel C4×Dic20 C20.44D4 C40⋊5C4 C4×C40 C4×Dic10 C2×Dic20 Dic20 C2×C20 C4×C8 C20 C20 C42 C2×C8 C10 C8 C2×C4 C4 C4 C2 # reps 1 2 1 1 2 1 8 2 2 4 2 2 4 4 8 8 16 8 16

Matrix representation of C4×Dic20 in GL4(𝔽41) generated by

 9 0 0 0 0 9 0 0 0 0 1 0 0 0 0 1
,
 40 1 0 0 5 35 0 0 0 0 15 3 0 0 38 35
,
 25 30 0 0 12 16 0 0 0 0 17 38 0 0 1 24
G:=sub<GL(4,GF(41))| [9,0,0,0,0,9,0,0,0,0,1,0,0,0,0,1],[40,5,0,0,1,35,0,0,0,0,15,38,0,0,3,35],[25,12,0,0,30,16,0,0,0,0,17,1,0,0,38,24] >;

C4×Dic20 in GAP, Magma, Sage, TeX

C_4\times {\rm Dic}_{20}
% in TeX

G:=Group("C4xDic20");
// GroupNames label

G:=SmallGroup(320,325);
// by ID

G=gap.SmallGroup(320,325);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,253,344,58,1684,102,12550]);
// Polycyclic

G:=Group<a,b,c|a^4=b^40=1,c^2=b^20,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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