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## G = C2×C4×C36order 288 = 25·32

### Abelian group of type [2,4,36]

Aliases: C2×C4×C36, SmallGroup(288,164)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C4×C36
 Chief series C1 — C3 — C6 — C2×C6 — C2×C18 — C2×C36 — C4×C36 — C2×C4×C36
 Lower central C1 — C2×C4×C36
 Upper central C1 — C2×C4×C36

Generators and relations for C2×C4×C36
G = < a,b,c | a2=b4=c36=1, ab=ba, ac=ca, bc=cb >

Subgroups: 162, all normal (12 characteristic)
C1, C2 [×7], C3, C4 [×12], C22, C22 [×6], C6 [×7], C2×C4 [×18], C23, C9, C12 [×12], C2×C6, C2×C6 [×6], C42 [×4], C22×C4 [×3], C18 [×7], C2×C12 [×18], C22×C6, C2×C42, C36 [×12], C2×C18, C2×C18 [×6], C4×C12 [×4], C22×C12 [×3], C2×C36 [×18], C22×C18, C2×C4×C12, C4×C36 [×4], C22×C36 [×3], C2×C4×C36
Quotients: C1, C2 [×7], C3, C4 [×12], C22 [×7], C6 [×7], C2×C4 [×18], C23, C9, C12 [×12], C2×C6 [×7], C42 [×4], C22×C4 [×3], C18 [×7], C2×C12 [×18], C22×C6, C2×C42, C36 [×12], C2×C18 [×7], C4×C12 [×4], C22×C12 [×3], C2×C36 [×18], C22×C18, C2×C4×C12, C4×C36 [×4], C22×C36 [×3], C2×C4×C36

Smallest permutation representation of C2×C4×C36
Regular action on 288 points
Generators in S288
(1 74)(2 75)(3 76)(4 77)(5 78)(6 79)(7 80)(8 81)(9 82)(10 83)(11 84)(12 85)(13 86)(14 87)(15 88)(16 89)(17 90)(18 91)(19 92)(20 93)(21 94)(22 95)(23 96)(24 97)(25 98)(26 99)(27 100)(28 101)(29 102)(30 103)(31 104)(32 105)(33 106)(34 107)(35 108)(36 73)(37 245)(38 246)(39 247)(40 248)(41 249)(42 250)(43 251)(44 252)(45 217)(46 218)(47 219)(48 220)(49 221)(50 222)(51 223)(52 224)(53 225)(54 226)(55 227)(56 228)(57 229)(58 230)(59 231)(60 232)(61 233)(62 234)(63 235)(64 236)(65 237)(66 238)(67 239)(68 240)(69 241)(70 242)(71 243)(72 244)(109 271)(110 272)(111 273)(112 274)(113 275)(114 276)(115 277)(116 278)(117 279)(118 280)(119 281)(120 282)(121 283)(122 284)(123 285)(124 286)(125 287)(126 288)(127 253)(128 254)(129 255)(130 256)(131 257)(132 258)(133 259)(134 260)(135 261)(136 262)(137 263)(138 264)(139 265)(140 266)(141 267)(142 268)(143 269)(144 270)(145 196)(146 197)(147 198)(148 199)(149 200)(150 201)(151 202)(152 203)(153 204)(154 205)(155 206)(156 207)(157 208)(158 209)(159 210)(160 211)(161 212)(162 213)(163 214)(164 215)(165 216)(166 181)(167 182)(168 183)(169 184)(170 185)(171 186)(172 187)(173 188)(174 189)(175 190)(176 191)(177 192)(178 193)(179 194)(180 195)
(1 219 164 286)(2 220 165 287)(3 221 166 288)(4 222 167 253)(5 223 168 254)(6 224 169 255)(7 225 170 256)(8 226 171 257)(9 227 172 258)(10 228 173 259)(11 229 174 260)(12 230 175 261)(13 231 176 262)(14 232 177 263)(15 233 178 264)(16 234 179 265)(17 235 180 266)(18 236 145 267)(19 237 146 268)(20 238 147 269)(21 239 148 270)(22 240 149 271)(23 241 150 272)(24 242 151 273)(25 243 152 274)(26 244 153 275)(27 245 154 276)(28 246 155 277)(29 247 156 278)(30 248 157 279)(31 249 158 280)(32 250 159 281)(33 251 160 282)(34 252 161 283)(35 217 162 284)(36 218 163 285)(37 205 114 100)(38 206 115 101)(39 207 116 102)(40 208 117 103)(41 209 118 104)(42 210 119 105)(43 211 120 106)(44 212 121 107)(45 213 122 108)(46 214 123 73)(47 215 124 74)(48 216 125 75)(49 181 126 76)(50 182 127 77)(51 183 128 78)(52 184 129 79)(53 185 130 80)(54 186 131 81)(55 187 132 82)(56 188 133 83)(57 189 134 84)(58 190 135 85)(59 191 136 86)(60 192 137 87)(61 193 138 88)(62 194 139 89)(63 195 140 90)(64 196 141 91)(65 197 142 92)(66 198 143 93)(67 199 144 94)(68 200 109 95)(69 201 110 96)(70 202 111 97)(71 203 112 98)(72 204 113 99)
(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)

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

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

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

288 conjugacy classes

 class 1 2A ··· 2G 3A 3B 4A ··· 4X 6A ··· 6N 9A ··· 9F 12A ··· 12AV 18A ··· 18AP 36A ··· 36EN order 1 2 ··· 2 3 3 4 ··· 4 6 ··· 6 9 ··· 9 12 ··· 12 18 ··· 18 36 ··· 36 size 1 1 ··· 1 1 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1

288 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 type + + + image C1 C2 C2 C3 C4 C6 C6 C9 C12 C18 C18 C36 kernel C2×C4×C36 C4×C36 C22×C36 C2×C4×C12 C2×C36 C4×C12 C22×C12 C2×C42 C2×C12 C42 C22×C4 C2×C4 # reps 1 4 3 2 24 8 6 6 48 24 18 144

Matrix representation of C2×C4×C36 in GL3(𝔽37) generated by

 36 0 0 0 1 0 0 0 1
,
 6 0 0 0 6 0 0 0 6
,
 29 0 0 0 5 0 0 0 33
G:=sub<GL(3,GF(37))| [36,0,0,0,1,0,0,0,1],[6,0,0,0,6,0,0,0,6],[29,0,0,0,5,0,0,0,33] >;

C2×C4×C36 in GAP, Magma, Sage, TeX

C_2\times C_4\times C_{36}
% in TeX

G:=Group("C2xC4xC36");
// GroupNames label

G:=SmallGroup(288,164);
// by ID

G=gap.SmallGroup(288,164);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,168,344,360]);
// Polycyclic

G:=Group<a,b,c|a^2=b^4=c^36=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations

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