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G = C2×Dic36order 288 = 25·32

Direct product of C2 and Dic36

Series: Derived Chief Lower central Upper central

 Derived series C1 — C36 — C2×Dic36
 Chief series C1 — C3 — C9 — C18 — C36 — Dic18 — C2×Dic18 — C2×Dic36
 Lower central C9 — C18 — C36 — C2×Dic36
 Upper central C1 — C22 — C2×C4 — C2×C8

Generators and relations for C2×Dic36
G = < a,b,c | a2=b72=1, c2=b36, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 384 in 90 conjugacy classes, 44 normal (22 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C6, C8, C2×C4, C2×C4, Q8, C9, Dic3, C12, C2×C6, C2×C8, Q16, C2×Q8, C18, C18, C24, Dic6, C2×Dic3, C2×C12, C2×Q16, Dic9, C36, C2×C18, Dic12, C2×C24, C2×Dic6, C72, Dic18, Dic18, C2×Dic9, C2×C36, C2×Dic12, Dic36, C2×C72, C2×Dic18, C2×Dic36
Quotients: C1, C2, C22, S3, D4, C23, D6, Q16, C2×D4, D9, D12, C22×S3, C2×Q16, D18, Dic12, C2×D12, D36, C22×D9, C2×Dic12, Dic36, C2×D36, C2×Dic36

Smallest permutation representation of C2×Dic36
Regular action on 288 points
Generators in S288
(1 282)(2 283)(3 284)(4 285)(5 286)(6 287)(7 288)(8 217)(9 218)(10 219)(11 220)(12 221)(13 222)(14 223)(15 224)(16 225)(17 226)(18 227)(19 228)(20 229)(21 230)(22 231)(23 232)(24 233)(25 234)(26 235)(27 236)(28 237)(29 238)(30 239)(31 240)(32 241)(33 242)(34 243)(35 244)(36 245)(37 246)(38 247)(39 248)(40 249)(41 250)(42 251)(43 252)(44 253)(45 254)(46 255)(47 256)(48 257)(49 258)(50 259)(51 260)(52 261)(53 262)(54 263)(55 264)(56 265)(57 266)(58 267)(59 268)(60 269)(61 270)(62 271)(63 272)(64 273)(65 274)(66 275)(67 276)(68 277)(69 278)(70 279)(71 280)(72 281)(73 201)(74 202)(75 203)(76 204)(77 205)(78 206)(79 207)(80 208)(81 209)(82 210)(83 211)(84 212)(85 213)(86 214)(87 215)(88 216)(89 145)(90 146)(91 147)(92 148)(93 149)(94 150)(95 151)(96 152)(97 153)(98 154)(99 155)(100 156)(101 157)(102 158)(103 159)(104 160)(105 161)(106 162)(107 163)(108 164)(109 165)(110 166)(111 167)(112 168)(113 169)(114 170)(115 171)(116 172)(117 173)(118 174)(119 175)(120 176)(121 177)(122 178)(123 179)(124 180)(125 181)(126 182)(127 183)(128 184)(129 185)(130 186)(131 187)(132 188)(133 189)(134 190)(135 191)(136 192)(137 193)(138 194)(139 195)(140 196)(141 197)(142 198)(143 199)(144 200)
(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)
(1 152 37 188)(2 151 38 187)(3 150 39 186)(4 149 40 185)(5 148 41 184)(6 147 42 183)(7 146 43 182)(8 145 44 181)(9 216 45 180)(10 215 46 179)(11 214 47 178)(12 213 48 177)(13 212 49 176)(14 211 50 175)(15 210 51 174)(16 209 52 173)(17 208 53 172)(18 207 54 171)(19 206 55 170)(20 205 56 169)(21 204 57 168)(22 203 58 167)(23 202 59 166)(24 201 60 165)(25 200 61 164)(26 199 62 163)(27 198 63 162)(28 197 64 161)(29 196 65 160)(30 195 66 159)(31 194 67 158)(32 193 68 157)(33 192 69 156)(34 191 70 155)(35 190 71 154)(36 189 72 153)(73 269 109 233)(74 268 110 232)(75 267 111 231)(76 266 112 230)(77 265 113 229)(78 264 114 228)(79 263 115 227)(80 262 116 226)(81 261 117 225)(82 260 118 224)(83 259 119 223)(84 258 120 222)(85 257 121 221)(86 256 122 220)(87 255 123 219)(88 254 124 218)(89 253 125 217)(90 252 126 288)(91 251 127 287)(92 250 128 286)(93 249 129 285)(94 248 130 284)(95 247 131 283)(96 246 132 282)(97 245 133 281)(98 244 134 280)(99 243 135 279)(100 242 136 278)(101 241 137 277)(102 240 138 276)(103 239 139 275)(104 238 140 274)(105 237 141 273)(106 236 142 272)(107 235 143 271)(108 234 144 270)

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

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

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

78 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 6A 6B 6C 8A 8B 8C 8D 9A 9B 9C 12A 12B 12C 12D 18A ··· 18I 24A ··· 24H 36A ··· 36L 72A ··· 72X order 1 2 2 2 3 4 4 4 4 4 4 6 6 6 8 8 8 8 9 9 9 12 12 12 12 18 ··· 18 24 ··· 24 36 ··· 36 72 ··· 72 size 1 1 1 1 2 2 2 36 36 36 36 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

78 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + - + + + + + - + + - image C1 C2 C2 C2 S3 D4 D4 D6 D6 Q16 D9 D12 D12 D18 D18 Dic12 D36 D36 Dic36 kernel C2×Dic36 Dic36 C2×C72 C2×Dic18 C2×C24 C36 C2×C18 C24 C2×C12 C18 C2×C8 C12 C2×C6 C8 C2×C4 C6 C4 C22 C2 # reps 1 4 1 2 1 1 1 2 1 4 3 2 2 6 3 8 6 6 24

Matrix representation of C2×Dic36 in GL4(𝔽73) generated by

 72 0 0 0 0 72 0 0 0 0 1 0 0 0 0 1
,
 28 70 0 0 3 31 0 0 0 0 41 51 0 0 22 63
,
 51 10 0 0 32 22 0 0 0 0 54 59 0 0 5 19
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,1,0,0,0,0,1],[28,3,0,0,70,31,0,0,0,0,41,22,0,0,51,63],[51,32,0,0,10,22,0,0,0,0,54,5,0,0,59,19] >;

C2×Dic36 in GAP, Magma, Sage, TeX

C_2\times {\rm Dic}_{36}
% in TeX

G:=Group("C2xDic36");
// GroupNames label

G:=SmallGroup(288,109);
// by ID

G=gap.SmallGroup(288,109);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,254,142,675,80,6725,292,9414]);
// Polycyclic

G:=Group<a,b,c|a^2=b^72=1,c^2=b^36,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

׿
×
𝔽