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## G = C2×C5⋊2C32order 320 = 26·5

### Direct product of C2 and C5⋊2C32

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C2×C5⋊2C32
 Chief series C1 — C5 — C10 — C20 — C40 — C80 — C5⋊2C32 — C2×C5⋊2C32
 Lower central C5 — C2×C5⋊2C32
 Upper central C1 — C2×C16

Generators and relations for C2×C52C32
G = < a,b,c | a2=b5=c32=1, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of C2×C52C32
Regular action on 320 points
Generators in S320
(1 125)(2 126)(3 127)(4 128)(5 97)(6 98)(7 99)(8 100)(9 101)(10 102)(11 103)(12 104)(13 105)(14 106)(15 107)(16 108)(17 109)(18 110)(19 111)(20 112)(21 113)(22 114)(23 115)(24 116)(25 117)(26 118)(27 119)(28 120)(29 121)(30 122)(31 123)(32 124)(33 237)(34 238)(35 239)(36 240)(37 241)(38 242)(39 243)(40 244)(41 245)(42 246)(43 247)(44 248)(45 249)(46 250)(47 251)(48 252)(49 253)(50 254)(51 255)(52 256)(53 225)(54 226)(55 227)(56 228)(57 229)(58 230)(59 231)(60 232)(61 233)(62 234)(63 235)(64 236)(65 164)(66 165)(67 166)(68 167)(69 168)(70 169)(71 170)(72 171)(73 172)(74 173)(75 174)(76 175)(77 176)(78 177)(79 178)(80 179)(81 180)(82 181)(83 182)(84 183)(85 184)(86 185)(87 186)(88 187)(89 188)(90 189)(91 190)(92 191)(93 192)(94 161)(95 162)(96 163)(129 222)(130 223)(131 224)(132 193)(133 194)(134 195)(135 196)(136 197)(137 198)(138 199)(139 200)(140 201)(141 202)(142 203)(143 204)(144 205)(145 206)(146 207)(147 208)(148 209)(149 210)(150 211)(151 212)(152 213)(153 214)(154 215)(155 216)(156 217)(157 218)(158 219)(159 220)(160 221)(257 290)(258 291)(259 292)(260 293)(261 294)(262 295)(263 296)(264 297)(265 298)(266 299)(267 300)(268 301)(269 302)(270 303)(271 304)(272 305)(273 306)(274 307)(275 308)(276 309)(277 310)(278 311)(279 312)(280 313)(281 314)(282 315)(283 316)(284 317)(285 318)(286 319)(287 320)(288 289)
(1 65 317 216 45)(2 46 217 318 66)(3 67 319 218 47)(4 48 219 320 68)(5 69 289 220 49)(6 50 221 290 70)(7 71 291 222 51)(8 52 223 292 72)(9 73 293 224 53)(10 54 193 294 74)(11 75 295 194 55)(12 56 195 296 76)(13 77 297 196 57)(14 58 197 298 78)(15 79 299 198 59)(16 60 199 300 80)(17 81 301 200 61)(18 62 201 302 82)(19 83 303 202 63)(20 64 203 304 84)(21 85 305 204 33)(22 34 205 306 86)(23 87 307 206 35)(24 36 207 308 88)(25 89 309 208 37)(26 38 209 310 90)(27 91 311 210 39)(28 40 211 312 92)(29 93 313 212 41)(30 42 213 314 94)(31 95 315 214 43)(32 44 215 316 96)(97 168 288 159 253)(98 254 160 257 169)(99 170 258 129 255)(100 256 130 259 171)(101 172 260 131 225)(102 226 132 261 173)(103 174 262 133 227)(104 228 134 263 175)(105 176 264 135 229)(106 230 136 265 177)(107 178 266 137 231)(108 232 138 267 179)(109 180 268 139 233)(110 234 140 269 181)(111 182 270 141 235)(112 236 142 271 183)(113 184 272 143 237)(114 238 144 273 185)(115 186 274 145 239)(116 240 146 275 187)(117 188 276 147 241)(118 242 148 277 189)(119 190 278 149 243)(120 244 150 279 191)(121 192 280 151 245)(122 246 152 281 161)(123 162 282 153 247)(124 248 154 283 163)(125 164 284 155 249)(126 250 156 285 165)(127 166 286 157 251)(128 252 158 287 167)
(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)| (1,125)(2,126)(3,127)(4,128)(5,97)(6,98)(7,99)(8,100)(9,101)(10,102)(11,103)(12,104)(13,105)(14,106)(15,107)(16,108)(17,109)(18,110)(19,111)(20,112)(21,113)(22,114)(23,115)(24,116)(25,117)(26,118)(27,119)(28,120)(29,121)(30,122)(31,123)(32,124)(33,237)(34,238)(35,239)(36,240)(37,241)(38,242)(39,243)(40,244)(41,245)(42,246)(43,247)(44,248)(45,249)(46,250)(47,251)(48,252)(49,253)(50,254)(51,255)(52,256)(53,225)(54,226)(55,227)(56,228)(57,229)(58,230)(59,231)(60,232)(61,233)(62,234)(63,235)(64,236)(65,164)(66,165)(67,166)(68,167)(69,168)(70,169)(71,170)(72,171)(73,172)(74,173)(75,174)(76,175)(77,176)(78,177)(79,178)(80,179)(81,180)(82,181)(83,182)(84,183)(85,184)(86,185)(87,186)(88,187)(89,188)(90,189)(91,190)(92,191)(93,192)(94,161)(95,162)(96,163)(129,222)(130,223)(131,224)(132,193)(133,194)(134,195)(135,196)(136,197)(137,198)(138,199)(139,200)(140,201)(141,202)(142,203)(143,204)(144,205)(145,206)(146,207)(147,208)(148,209)(149,210)(150,211)(151,212)(152,213)(153,214)(154,215)(155,216)(156,217)(157,218)(158,219)(159,220)(160,221)(257,290)(258,291)(259,292)(260,293)(261,294)(262,295)(263,296)(264,297)(265,298)(266,299)(267,300)(268,301)(269,302)(270,303)(271,304)(272,305)(273,306)(274,307)(275,308)(276,309)(277,310)(278,311)(279,312)(280,313)(281,314)(282,315)(283,316)(284,317)(285,318)(286,319)(287,320)(288,289), (1,65,317,216,45)(2,46,217,318,66)(3,67,319,218,47)(4,48,219,320,68)(5,69,289,220,49)(6,50,221,290,70)(7,71,291,222,51)(8,52,223,292,72)(9,73,293,224,53)(10,54,193,294,74)(11,75,295,194,55)(12,56,195,296,76)(13,77,297,196,57)(14,58,197,298,78)(15,79,299,198,59)(16,60,199,300,80)(17,81,301,200,61)(18,62,201,302,82)(19,83,303,202,63)(20,64,203,304,84)(21,85,305,204,33)(22,34,205,306,86)(23,87,307,206,35)(24,36,207,308,88)(25,89,309,208,37)(26,38,209,310,90)(27,91,311,210,39)(28,40,211,312,92)(29,93,313,212,41)(30,42,213,314,94)(31,95,315,214,43)(32,44,215,316,96)(97,168,288,159,253)(98,254,160,257,169)(99,170,258,129,255)(100,256,130,259,171)(101,172,260,131,225)(102,226,132,261,173)(103,174,262,133,227)(104,228,134,263,175)(105,176,264,135,229)(106,230,136,265,177)(107,178,266,137,231)(108,232,138,267,179)(109,180,268,139,233)(110,234,140,269,181)(111,182,270,141,235)(112,236,142,271,183)(113,184,272,143,237)(114,238,144,273,185)(115,186,274,145,239)(116,240,146,275,187)(117,188,276,147,241)(118,242,148,277,189)(119,190,278,149,243)(120,244,150,279,191)(121,192,280,151,245)(122,246,152,281,161)(123,162,282,153,247)(124,248,154,283,163)(125,164,284,155,249)(126,250,156,285,165)(127,166,286,157,251)(128,252,158,287,167), (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( (1,125)(2,126)(3,127)(4,128)(5,97)(6,98)(7,99)(8,100)(9,101)(10,102)(11,103)(12,104)(13,105)(14,106)(15,107)(16,108)(17,109)(18,110)(19,111)(20,112)(21,113)(22,114)(23,115)(24,116)(25,117)(26,118)(27,119)(28,120)(29,121)(30,122)(31,123)(32,124)(33,237)(34,238)(35,239)(36,240)(37,241)(38,242)(39,243)(40,244)(41,245)(42,246)(43,247)(44,248)(45,249)(46,250)(47,251)(48,252)(49,253)(50,254)(51,255)(52,256)(53,225)(54,226)(55,227)(56,228)(57,229)(58,230)(59,231)(60,232)(61,233)(62,234)(63,235)(64,236)(65,164)(66,165)(67,166)(68,167)(69,168)(70,169)(71,170)(72,171)(73,172)(74,173)(75,174)(76,175)(77,176)(78,177)(79,178)(80,179)(81,180)(82,181)(83,182)(84,183)(85,184)(86,185)(87,186)(88,187)(89,188)(90,189)(91,190)(92,191)(93,192)(94,161)(95,162)(96,163)(129,222)(130,223)(131,224)(132,193)(133,194)(134,195)(135,196)(136,197)(137,198)(138,199)(139,200)(140,201)(141,202)(142,203)(143,204)(144,205)(145,206)(146,207)(147,208)(148,209)(149,210)(150,211)(151,212)(152,213)(153,214)(154,215)(155,216)(156,217)(157,218)(158,219)(159,220)(160,221)(257,290)(258,291)(259,292)(260,293)(261,294)(262,295)(263,296)(264,297)(265,298)(266,299)(267,300)(268,301)(269,302)(270,303)(271,304)(272,305)(273,306)(274,307)(275,308)(276,309)(277,310)(278,311)(279,312)(280,313)(281,314)(282,315)(283,316)(284,317)(285,318)(286,319)(287,320)(288,289), (1,65,317,216,45)(2,46,217,318,66)(3,67,319,218,47)(4,48,219,320,68)(5,69,289,220,49)(6,50,221,290,70)(7,71,291,222,51)(8,52,223,292,72)(9,73,293,224,53)(10,54,193,294,74)(11,75,295,194,55)(12,56,195,296,76)(13,77,297,196,57)(14,58,197,298,78)(15,79,299,198,59)(16,60,199,300,80)(17,81,301,200,61)(18,62,201,302,82)(19,83,303,202,63)(20,64,203,304,84)(21,85,305,204,33)(22,34,205,306,86)(23,87,307,206,35)(24,36,207,308,88)(25,89,309,208,37)(26,38,209,310,90)(27,91,311,210,39)(28,40,211,312,92)(29,93,313,212,41)(30,42,213,314,94)(31,95,315,214,43)(32,44,215,316,96)(97,168,288,159,253)(98,254,160,257,169)(99,170,258,129,255)(100,256,130,259,171)(101,172,260,131,225)(102,226,132,261,173)(103,174,262,133,227)(104,228,134,263,175)(105,176,264,135,229)(106,230,136,265,177)(107,178,266,137,231)(108,232,138,267,179)(109,180,268,139,233)(110,234,140,269,181)(111,182,270,141,235)(112,236,142,271,183)(113,184,272,143,237)(114,238,144,273,185)(115,186,274,145,239)(116,240,146,275,187)(117,188,276,147,241)(118,242,148,277,189)(119,190,278,149,243)(120,244,150,279,191)(121,192,280,151,245)(122,246,152,281,161)(123,162,282,153,247)(124,248,154,283,163)(125,164,284,155,249)(126,250,156,285,165)(127,166,286,157,251)(128,252,158,287,167), (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([[(1,125),(2,126),(3,127),(4,128),(5,97),(6,98),(7,99),(8,100),(9,101),(10,102),(11,103),(12,104),(13,105),(14,106),(15,107),(16,108),(17,109),(18,110),(19,111),(20,112),(21,113),(22,114),(23,115),(24,116),(25,117),(26,118),(27,119),(28,120),(29,121),(30,122),(31,123),(32,124),(33,237),(34,238),(35,239),(36,240),(37,241),(38,242),(39,243),(40,244),(41,245),(42,246),(43,247),(44,248),(45,249),(46,250),(47,251),(48,252),(49,253),(50,254),(51,255),(52,256),(53,225),(54,226),(55,227),(56,228),(57,229),(58,230),(59,231),(60,232),(61,233),(62,234),(63,235),(64,236),(65,164),(66,165),(67,166),(68,167),(69,168),(70,169),(71,170),(72,171),(73,172),(74,173),(75,174),(76,175),(77,176),(78,177),(79,178),(80,179),(81,180),(82,181),(83,182),(84,183),(85,184),(86,185),(87,186),(88,187),(89,188),(90,189),(91,190),(92,191),(93,192),(94,161),(95,162),(96,163),(129,222),(130,223),(131,224),(132,193),(133,194),(134,195),(135,196),(136,197),(137,198),(138,199),(139,200),(140,201),(141,202),(142,203),(143,204),(144,205),(145,206),(146,207),(147,208),(148,209),(149,210),(150,211),(151,212),(152,213),(153,214),(154,215),(155,216),(156,217),(157,218),(158,219),(159,220),(160,221),(257,290),(258,291),(259,292),(260,293),(261,294),(262,295),(263,296),(264,297),(265,298),(266,299),(267,300),(268,301),(269,302),(270,303),(271,304),(272,305),(273,306),(274,307),(275,308),(276,309),(277,310),(278,311),(279,312),(280,313),(281,314),(282,315),(283,316),(284,317),(285,318),(286,319),(287,320),(288,289)], [(1,65,317,216,45),(2,46,217,318,66),(3,67,319,218,47),(4,48,219,320,68),(5,69,289,220,49),(6,50,221,290,70),(7,71,291,222,51),(8,52,223,292,72),(9,73,293,224,53),(10,54,193,294,74),(11,75,295,194,55),(12,56,195,296,76),(13,77,297,196,57),(14,58,197,298,78),(15,79,299,198,59),(16,60,199,300,80),(17,81,301,200,61),(18,62,201,302,82),(19,83,303,202,63),(20,64,203,304,84),(21,85,305,204,33),(22,34,205,306,86),(23,87,307,206,35),(24,36,207,308,88),(25,89,309,208,37),(26,38,209,310,90),(27,91,311,210,39),(28,40,211,312,92),(29,93,313,212,41),(30,42,213,314,94),(31,95,315,214,43),(32,44,215,316,96),(97,168,288,159,253),(98,254,160,257,169),(99,170,258,129,255),(100,256,130,259,171),(101,172,260,131,225),(102,226,132,261,173),(103,174,262,133,227),(104,228,134,263,175),(105,176,264,135,229),(106,230,136,265,177),(107,178,266,137,231),(108,232,138,267,179),(109,180,268,139,233),(110,234,140,269,181),(111,182,270,141,235),(112,236,142,271,183),(113,184,272,143,237),(114,238,144,273,185),(115,186,274,145,239),(116,240,146,275,187),(117,188,276,147,241),(118,242,148,277,189),(119,190,278,149,243),(120,244,150,279,191),(121,192,280,151,245),(122,246,152,281,161),(123,162,282,153,247),(124,248,154,283,163),(125,164,284,155,249),(126,250,156,285,165),(127,166,286,157,251),(128,252,158,287,167)], [(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)]])

128 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 5A 5B 8A ··· 8H 10A ··· 10F 16A ··· 16P 20A ··· 20H 32A ··· 32AF 40A ··· 40P 80A ··· 80AF order 1 2 2 2 4 4 4 4 5 5 8 ··· 8 10 ··· 10 16 ··· 16 20 ··· 20 32 ··· 32 40 ··· 40 80 ··· 80 size 1 1 1 1 1 1 1 1 2 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 5 ··· 5 2 ··· 2 2 ··· 2

128 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + - + - image C1 C2 C2 C4 C4 C8 C8 C16 C16 C32 D5 Dic5 D10 Dic5 C5⋊2C8 C5⋊2C8 C5⋊2C16 C5⋊2C16 C5⋊2C32 kernel C2×C5⋊2C32 C5⋊2C32 C2×C80 C80 C2×C40 C40 C2×C20 C20 C2×C10 C10 C2×C16 C16 C16 C2×C8 C8 C2×C4 C4 C22 C2 # reps 1 2 1 2 2 4 4 8 8 32 2 2 2 2 4 4 8 8 32

Matrix representation of C2×C52C32 in GL4(𝔽641) generated by

 640 0 0 0 0 640 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 362 640 0 0 1 0
,
 4 0 0 0 0 640 0 0 0 0 143 276 0 0 121 498
G:=sub<GL(4,GF(641))| [640,0,0,0,0,640,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,362,1,0,0,640,0],[4,0,0,0,0,640,0,0,0,0,143,121,0,0,276,498] >;

C2×C52C32 in GAP, Magma, Sage, TeX

C_2\times C_5\rtimes_2C_{32}
% in TeX

G:=Group("C2xC5:2C32");
// GroupNames label

G:=SmallGroup(320,56);
// by ID

G=gap.SmallGroup(320,56);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,58,80,102,12550]);
// Polycyclic

G:=Group<a,b,c|a^2=b^5=c^32=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

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