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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 [×2], C3, C4 [×2], C4 [×4], C22, C6, C6 [×2], C8 [×2], C2×C4, C2×C4 [×2], Q8 [×6], C9, Dic3 [×4], C12 [×2], C2×C6, C2×C8, Q16 [×4], C2×Q8 [×2], C18, C18 [×2], C24 [×2], Dic6 [×6], C2×Dic3 [×2], C2×C12, C2×Q16, Dic9 [×4], C36 [×2], C2×C18, Dic12 [×4], C2×C24, C2×Dic6 [×2], C72 [×2], Dic18 [×4], Dic18 [×2], C2×Dic9 [×2], C2×C36, C2×Dic12, Dic36 [×4], C2×C72, C2×Dic18 [×2], C2×Dic36
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], Q16 [×2], C2×D4, D9, D12 [×2], C22×S3, C2×Q16, D18 [×3], Dic12 [×2], C2×D12, D36 [×2], C22×D9, C2×Dic12, Dic36 [×2], C2×D36, C2×Dic36

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

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

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

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

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