direct product, abelian, monomial, 2-elementary
Aliases: C22×C80, SmallGroup(320,1003)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C22×C80 |
| C1 — C22×C80 |
| C1 — C22×C80 |
Subgroups: 98, all normal (16 characteristic)
C1, C2, C2 [×6], C4, C4 [×3], C22 [×7], C5, C8, C8 [×3], C2×C4 [×6], C23, C10, C10 [×6], C16 [×4], C2×C8 [×6], C22×C4, C20, C20 [×3], C2×C10 [×7], C2×C16 [×6], C22×C8, C40, C40 [×3], C2×C20 [×6], C22×C10, C22×C16, C80 [×4], C2×C40 [×6], C22×C20, C2×C80 [×6], C22×C40, C22×C80
Quotients:
C1, C2 [×7], C4 [×4], C22 [×7], C5, C8 [×4], C2×C4 [×6], C23, C10 [×7], C16 [×4], C2×C8 [×6], C22×C4, C20 [×4], C2×C10 [×7], C2×C16 [×6], C22×C8, C40 [×4], C2×C20 [×6], C22×C10, C22×C16, C80 [×4], C2×C40 [×6], C22×C20, C2×C80 [×6], C22×C40, C22×C80
Generators and relations
G = < a,b,c | a2=b2=c80=1, ab=ba, ac=ca, bc=cb >
►Regular action on 320 points
Generators in S320
(1 268)(2 269)(3 270)(4 271)(5 272)(6 273)(7 274)(8 275)(9 276)(10 277)(11 278)(12 279)(13 280)(14 281)(15 282)(16 283)(17 284)(18 285)(19 286)(20 287)(21 288)(22 289)(23 290)(24 291)(25 292)(26 293)(27 294)(28 295)(29 296)(30 297)(31 298)(32 299)(33 300)(34 301)(35 302)(36 303)(37 304)(38 305)(39 306)(40 307)(41 308)(42 309)(43 310)(44 311)(45 312)(46 313)(47 314)(48 315)(49 316)(50 317)(51 318)(52 319)(53 320)(54 241)(55 242)(56 243)(57 244)(58 245)(59 246)(60 247)(61 248)(62 249)(63 250)(64 251)(65 252)(66 253)(67 254)(68 255)(69 256)(70 257)(71 258)(72 259)(73 260)(74 261)(75 262)(76 263)(77 264)(78 265)(79 266)(80 267)(81 191)(82 192)(83 193)(84 194)(85 195)(86 196)(87 197)(88 198)(89 199)(90 200)(91 201)(92 202)(93 203)(94 204)(95 205)(96 206)(97 207)(98 208)(99 209)(100 210)(101 211)(102 212)(103 213)(104 214)(105 215)(106 216)(107 217)(108 218)(109 219)(110 220)(111 221)(112 222)(113 223)(114 224)(115 225)(116 226)(117 227)(118 228)(119 229)(120 230)(121 231)(122 232)(123 233)(124 234)(125 235)(126 236)(127 237)(128 238)(129 239)(130 240)(131 161)(132 162)(133 163)(134 164)(135 165)(136 166)(137 167)(138 168)(139 169)(140 170)(141 171)(142 172)(143 173)(144 174)(145 175)(146 176)(147 177)(148 178)(149 179)(150 180)(151 181)(152 182)(153 183)(154 184)(155 185)(156 186)(157 187)(158 188)(159 189)(160 190)
(1 89)(2 90)(3 91)(4 92)(5 93)(6 94)(7 95)(8 96)(9 97)(10 98)(11 99)(12 100)(13 101)(14 102)(15 103)(16 104)(17 105)(18 106)(19 107)(20 108)(21 109)(22 110)(23 111)(24 112)(25 113)(26 114)(27 115)(28 116)(29 117)(30 118)(31 119)(32 120)(33 121)(34 122)(35 123)(36 124)(37 125)(38 126)(39 127)(40 128)(41 129)(42 130)(43 131)(44 132)(45 133)(46 134)(47 135)(48 136)(49 137)(50 138)(51 139)(52 140)(53 141)(54 142)(55 143)(56 144)(57 145)(58 146)(59 147)(60 148)(61 149)(62 150)(63 151)(64 152)(65 153)(66 154)(67 155)(68 156)(69 157)(70 158)(71 159)(72 160)(73 81)(74 82)(75 83)(76 84)(77 85)(78 86)(79 87)(80 88)(161 310)(162 311)(163 312)(164 313)(165 314)(166 315)(167 316)(168 317)(169 318)(170 319)(171 320)(172 241)(173 242)(174 243)(175 244)(176 245)(177 246)(178 247)(179 248)(180 249)(181 250)(182 251)(183 252)(184 253)(185 254)(186 255)(187 256)(188 257)(189 258)(190 259)(191 260)(192 261)(193 262)(194 263)(195 264)(196 265)(197 266)(198 267)(199 268)(200 269)(201 270)(202 271)(203 272)(204 273)(205 274)(206 275)(207 276)(208 277)(209 278)(210 279)(211 280)(212 281)(213 282)(214 283)(215 284)(216 285)(217 286)(218 287)(219 288)(220 289)(221 290)(222 291)(223 292)(224 293)(225 294)(226 295)(227 296)(228 297)(229 298)(230 299)(231 300)(232 301)(233 302)(234 303)(235 304)(236 305)(237 306)(238 307)(239 308)(240 309)
(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)| (1,268)(2,269)(3,270)(4,271)(5,272)(6,273)(7,274)(8,275)(9,276)(10,277)(11,278)(12,279)(13,280)(14,281)(15,282)(16,283)(17,284)(18,285)(19,286)(20,287)(21,288)(22,289)(23,290)(24,291)(25,292)(26,293)(27,294)(28,295)(29,296)(30,297)(31,298)(32,299)(33,300)(34,301)(35,302)(36,303)(37,304)(38,305)(39,306)(40,307)(41,308)(42,309)(43,310)(44,311)(45,312)(46,313)(47,314)(48,315)(49,316)(50,317)(51,318)(52,319)(53,320)(54,241)(55,242)(56,243)(57,244)(58,245)(59,246)(60,247)(61,248)(62,249)(63,250)(64,251)(65,252)(66,253)(67,254)(68,255)(69,256)(70,257)(71,258)(72,259)(73,260)(74,261)(75,262)(76,263)(77,264)(78,265)(79,266)(80,267)(81,191)(82,192)(83,193)(84,194)(85,195)(86,196)(87,197)(88,198)(89,199)(90,200)(91,201)(92,202)(93,203)(94,204)(95,205)(96,206)(97,207)(98,208)(99,209)(100,210)(101,211)(102,212)(103,213)(104,214)(105,215)(106,216)(107,217)(108,218)(109,219)(110,220)(111,221)(112,222)(113,223)(114,224)(115,225)(116,226)(117,227)(118,228)(119,229)(120,230)(121,231)(122,232)(123,233)(124,234)(125,235)(126,236)(127,237)(128,238)(129,239)(130,240)(131,161)(132,162)(133,163)(134,164)(135,165)(136,166)(137,167)(138,168)(139,169)(140,170)(141,171)(142,172)(143,173)(144,174)(145,175)(146,176)(147,177)(148,178)(149,179)(150,180)(151,181)(152,182)(153,183)(154,184)(155,185)(156,186)(157,187)(158,188)(159,189)(160,190), (1,89)(2,90)(3,91)(4,92)(5,93)(6,94)(7,95)(8,96)(9,97)(10,98)(11,99)(12,100)(13,101)(14,102)(15,103)(16,104)(17,105)(18,106)(19,107)(20,108)(21,109)(22,110)(23,111)(24,112)(25,113)(26,114)(27,115)(28,116)(29,117)(30,118)(31,119)(32,120)(33,121)(34,122)(35,123)(36,124)(37,125)(38,126)(39,127)(40,128)(41,129)(42,130)(43,131)(44,132)(45,133)(46,134)(47,135)(48,136)(49,137)(50,138)(51,139)(52,140)(53,141)(54,142)(55,143)(56,144)(57,145)(58,146)(59,147)(60,148)(61,149)(62,150)(63,151)(64,152)(65,153)(66,154)(67,155)(68,156)(69,157)(70,158)(71,159)(72,160)(73,81)(74,82)(75,83)(76,84)(77,85)(78,86)(79,87)(80,88)(161,310)(162,311)(163,312)(164,313)(165,314)(166,315)(167,316)(168,317)(169,318)(170,319)(171,320)(172,241)(173,242)(174,243)(175,244)(176,245)(177,246)(178,247)(179,248)(180,249)(181,250)(182,251)(183,252)(184,253)(185,254)(186,255)(187,256)(188,257)(189,258)(190,259)(191,260)(192,261)(193,262)(194,263)(195,264)(196,265)(197,266)(198,267)(199,268)(200,269)(201,270)(202,271)(203,272)(204,273)(205,274)(206,275)(207,276)(208,277)(209,278)(210,279)(211,280)(212,281)(213,282)(214,283)(215,284)(216,285)(217,286)(218,287)(219,288)(220,289)(221,290)(222,291)(223,292)(224,293)(225,294)(226,295)(227,296)(228,297)(229,298)(230,299)(231,300)(232,301)(233,302)(234,303)(235,304)(236,305)(237,306)(238,307)(239,308)(240,309), (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( (1,268)(2,269)(3,270)(4,271)(5,272)(6,273)(7,274)(8,275)(9,276)(10,277)(11,278)(12,279)(13,280)(14,281)(15,282)(16,283)(17,284)(18,285)(19,286)(20,287)(21,288)(22,289)(23,290)(24,291)(25,292)(26,293)(27,294)(28,295)(29,296)(30,297)(31,298)(32,299)(33,300)(34,301)(35,302)(36,303)(37,304)(38,305)(39,306)(40,307)(41,308)(42,309)(43,310)(44,311)(45,312)(46,313)(47,314)(48,315)(49,316)(50,317)(51,318)(52,319)(53,320)(54,241)(55,242)(56,243)(57,244)(58,245)(59,246)(60,247)(61,248)(62,249)(63,250)(64,251)(65,252)(66,253)(67,254)(68,255)(69,256)(70,257)(71,258)(72,259)(73,260)(74,261)(75,262)(76,263)(77,264)(78,265)(79,266)(80,267)(81,191)(82,192)(83,193)(84,194)(85,195)(86,196)(87,197)(88,198)(89,199)(90,200)(91,201)(92,202)(93,203)(94,204)(95,205)(96,206)(97,207)(98,208)(99,209)(100,210)(101,211)(102,212)(103,213)(104,214)(105,215)(106,216)(107,217)(108,218)(109,219)(110,220)(111,221)(112,222)(113,223)(114,224)(115,225)(116,226)(117,227)(118,228)(119,229)(120,230)(121,231)(122,232)(123,233)(124,234)(125,235)(126,236)(127,237)(128,238)(129,239)(130,240)(131,161)(132,162)(133,163)(134,164)(135,165)(136,166)(137,167)(138,168)(139,169)(140,170)(141,171)(142,172)(143,173)(144,174)(145,175)(146,176)(147,177)(148,178)(149,179)(150,180)(151,181)(152,182)(153,183)(154,184)(155,185)(156,186)(157,187)(158,188)(159,189)(160,190), (1,89)(2,90)(3,91)(4,92)(5,93)(6,94)(7,95)(8,96)(9,97)(10,98)(11,99)(12,100)(13,101)(14,102)(15,103)(16,104)(17,105)(18,106)(19,107)(20,108)(21,109)(22,110)(23,111)(24,112)(25,113)(26,114)(27,115)(28,116)(29,117)(30,118)(31,119)(32,120)(33,121)(34,122)(35,123)(36,124)(37,125)(38,126)(39,127)(40,128)(41,129)(42,130)(43,131)(44,132)(45,133)(46,134)(47,135)(48,136)(49,137)(50,138)(51,139)(52,140)(53,141)(54,142)(55,143)(56,144)(57,145)(58,146)(59,147)(60,148)(61,149)(62,150)(63,151)(64,152)(65,153)(66,154)(67,155)(68,156)(69,157)(70,158)(71,159)(72,160)(73,81)(74,82)(75,83)(76,84)(77,85)(78,86)(79,87)(80,88)(161,310)(162,311)(163,312)(164,313)(165,314)(166,315)(167,316)(168,317)(169,318)(170,319)(171,320)(172,241)(173,242)(174,243)(175,244)(176,245)(177,246)(178,247)(179,248)(180,249)(181,250)(182,251)(183,252)(184,253)(185,254)(186,255)(187,256)(188,257)(189,258)(190,259)(191,260)(192,261)(193,262)(194,263)(195,264)(196,265)(197,266)(198,267)(199,268)(200,269)(201,270)(202,271)(203,272)(204,273)(205,274)(206,275)(207,276)(208,277)(209,278)(210,279)(211,280)(212,281)(213,282)(214,283)(215,284)(216,285)(217,286)(218,287)(219,288)(220,289)(221,290)(222,291)(223,292)(224,293)(225,294)(226,295)(227,296)(228,297)(229,298)(230,299)(231,300)(232,301)(233,302)(234,303)(235,304)(236,305)(237,306)(238,307)(239,308)(240,309), (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([(1,268),(2,269),(3,270),(4,271),(5,272),(6,273),(7,274),(8,275),(9,276),(10,277),(11,278),(12,279),(13,280),(14,281),(15,282),(16,283),(17,284),(18,285),(19,286),(20,287),(21,288),(22,289),(23,290),(24,291),(25,292),(26,293),(27,294),(28,295),(29,296),(30,297),(31,298),(32,299),(33,300),(34,301),(35,302),(36,303),(37,304),(38,305),(39,306),(40,307),(41,308),(42,309),(43,310),(44,311),(45,312),(46,313),(47,314),(48,315),(49,316),(50,317),(51,318),(52,319),(53,320),(54,241),(55,242),(56,243),(57,244),(58,245),(59,246),(60,247),(61,248),(62,249),(63,250),(64,251),(65,252),(66,253),(67,254),(68,255),(69,256),(70,257),(71,258),(72,259),(73,260),(74,261),(75,262),(76,263),(77,264),(78,265),(79,266),(80,267),(81,191),(82,192),(83,193),(84,194),(85,195),(86,196),(87,197),(88,198),(89,199),(90,200),(91,201),(92,202),(93,203),(94,204),(95,205),(96,206),(97,207),(98,208),(99,209),(100,210),(101,211),(102,212),(103,213),(104,214),(105,215),(106,216),(107,217),(108,218),(109,219),(110,220),(111,221),(112,222),(113,223),(114,224),(115,225),(116,226),(117,227),(118,228),(119,229),(120,230),(121,231),(122,232),(123,233),(124,234),(125,235),(126,236),(127,237),(128,238),(129,239),(130,240),(131,161),(132,162),(133,163),(134,164),(135,165),(136,166),(137,167),(138,168),(139,169),(140,170),(141,171),(142,172),(143,173),(144,174),(145,175),(146,176),(147,177),(148,178),(149,179),(150,180),(151,181),(152,182),(153,183),(154,184),(155,185),(156,186),(157,187),(158,188),(159,189),(160,190)], [(1,89),(2,90),(3,91),(4,92),(5,93),(6,94),(7,95),(8,96),(9,97),(10,98),(11,99),(12,100),(13,101),(14,102),(15,103),(16,104),(17,105),(18,106),(19,107),(20,108),(21,109),(22,110),(23,111),(24,112),(25,113),(26,114),(27,115),(28,116),(29,117),(30,118),(31,119),(32,120),(33,121),(34,122),(35,123),(36,124),(37,125),(38,126),(39,127),(40,128),(41,129),(42,130),(43,131),(44,132),(45,133),(46,134),(47,135),(48,136),(49,137),(50,138),(51,139),(52,140),(53,141),(54,142),(55,143),(56,144),(57,145),(58,146),(59,147),(60,148),(61,149),(62,150),(63,151),(64,152),(65,153),(66,154),(67,155),(68,156),(69,157),(70,158),(71,159),(72,160),(73,81),(74,82),(75,83),(76,84),(77,85),(78,86),(79,87),(80,88),(161,310),(162,311),(163,312),(164,313),(165,314),(166,315),(167,316),(168,317),(169,318),(170,319),(171,320),(172,241),(173,242),(174,243),(175,244),(176,245),(177,246),(178,247),(179,248),(180,249),(181,250),(182,251),(183,252),(184,253),(185,254),(186,255),(187,256),(188,257),(189,258),(190,259),(191,260),(192,261),(193,262),(194,263),(195,264),(196,265),(197,266),(198,267),(199,268),(200,269),(201,270),(202,271),(203,272),(204,273),(205,274),(206,275),(207,276),(208,277),(209,278),(210,279),(211,280),(212,281),(213,282),(214,283),(215,284),(216,285),(217,286),(218,287),(219,288),(220,289),(221,290),(222,291),(223,292),(224,293),(225,294),(226,295),(227,296),(228,297),(229,298),(230,299),(231,300),(232,301),(233,302),(234,303),(235,304),(236,305),(237,306),(238,307),(239,308),(240,309)], [(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)])
Matrix representation ►G ⊆ GL3(𝔽241) generated by
| 1 | 0 | 0 |
| 0 | 240 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 240 | 0 |
| 0 | 0 | 240 |
| 111 | 0 | 0 |
| 0 | 136 | 0 |
| 0 | 0 | 73 |
G:=sub<GL(3,GF(241))| [1,0,0,0,240,0,0,0,1],[1,0,0,0,240,0,0,0,240],[111,0,0,0,136,0,0,0,73] >;
320 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4H | 5A | 5B | 5C | 5D | 8A | ··· | 8P | 10A | ··· | 10AB | 16A | ··· | 16AF | 20A | ··· | 20AF | 40A | ··· | 40BL | 80A | ··· | 80DX |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 16 | ··· | 16 | 20 | ··· | 20 | 40 | ··· | 40 | 80 | ··· | 80 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
320 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | + | |||||||||||||
| image | C1 | C2 | C2 | C4 | C4 | C5 | C8 | C8 | C10 | C10 | C16 | C20 | C20 | C40 | C40 | C80 |
| kernel | C22×C80 | C2×C80 | C22×C40 | C2×C40 | C22×C20 | C22×C16 | C2×C20 | C22×C10 | C2×C16 | C22×C8 | C2×C10 | C2×C8 | C22×C4 | C2×C4 | C23 | C22 |
| # reps | 1 | 6 | 1 | 6 | 2 | 4 | 12 | 4 | 24 | 4 | 32 | 24 | 8 | 48 | 16 | 128 |
In GAP, Magma, Sage, TeX
C_2^2\times C_{80} % in TeX
G:=Group("C2^2xC80"); // GroupNames label
G:=SmallGroup(320,1003);
// by ID
G=gap.SmallGroup(320,1003);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,280,102,124]);
// Polycyclic
G:=Group<a,b,c|a^2=b^2=c^80=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations
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