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G = C18.C42order 288 = 25·32

5th non-split extension by C18 of C42 acting via C42/C2×C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C18.5C42, C22.11D36, C23.33D18, C22.3Dic18, (C2×C36)⋊3C4, (C2×C4)⋊2Dic9, C18.9(C4⋊C4), (C2×C18).4Q8, (C2×Dic9)⋊2C4, (C2×C6).24D12, (C2×C18).33D4, C9⋊(C2.C42), C2.5(C4×Dic9), (C22×C4).4D9, C6.18(D6⋊C4), C2.2(D18⋊C4), (C22×C36).1C2, (C22×C12).9S3, C2.2(C4⋊Dic9), C6.10(C4×Dic3), (C2×C12).8Dic3, (C2×C6).13Dic6, C22.12(C4×D9), C2.2(Dic9⋊C4), C6.10(C4⋊Dic3), (C22×C6).133D6, C3.(C6.C42), C18.11(C22⋊C4), C6.14(Dic3⋊C4), C22.16(C9⋊D4), (C22×Dic9).1C2, C22.10(C2×Dic9), C2.2(C18.D4), C6.13(C6.D4), (C22×C18).31C22, (C2×C6).38(C4×S3), (C2×C18).28(C2×C4), (C2×C6).71(C3⋊D4), (C2×C6).32(C2×Dic3), SmallGroup(288,38)

Series: Derived Chief Lower central Upper central

C1C18 — C18.C42
C1C3C9C18C2×C18C22×C18C22×Dic9 — C18.C42
C9C18 — C18.C42
C1C23C22×C4

Generators and relations for C18.C42
 G = < a,b,c | a18=b4=c4=1, bab-1=a-1, ac=ca, cbc-1=a9b >

Subgroups: 380 in 114 conjugacy classes, 64 normal (28 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×6], C22 [×3], C22 [×4], C6 [×3], C6 [×4], C2×C4 [×2], C2×C4 [×10], C23, C9, Dic3 [×4], C12 [×2], C2×C6 [×3], C2×C6 [×4], C22×C4, C22×C4 [×2], C18 [×3], C18 [×4], C2×Dic3 [×8], C2×C12 [×2], C2×C12 [×2], C22×C6, C2.C42, Dic9 [×4], C36 [×2], C2×C18 [×3], C2×C18 [×4], C22×Dic3 [×2], C22×C12, C2×Dic9 [×4], C2×Dic9 [×4], C2×C36 [×2], C2×C36 [×2], C22×C18, C6.C42, C22×Dic9 [×2], C22×C36, C18.C42
Quotients: C1, C2 [×3], C4 [×6], C22, S3, C2×C4 [×3], D4 [×3], Q8, Dic3 [×2], D6, C42, C22⋊C4 [×3], C4⋊C4 [×3], D9, Dic6, C4×S3 [×2], D12, C2×Dic3, C3⋊D4 [×2], C2.C42, Dic9 [×2], D18, C4×Dic3, Dic3⋊C4 [×2], C4⋊Dic3, D6⋊C4 [×2], C6.D4, Dic18, C4×D9 [×2], D36, C2×Dic9, C9⋊D4 [×2], C6.C42, C4×Dic9, Dic9⋊C4 [×2], C4⋊Dic9, D18⋊C4 [×2], C18.D4, C18.C42

Smallest permutation representation of C18.C42
Regular action on 288 points
Generators in S288
(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)
(1 188 50 154)(2 187 51 153)(3 186 52 152)(4 185 53 151)(5 184 54 150)(6 183 37 149)(7 182 38 148)(8 181 39 147)(9 198 40 146)(10 197 41 145)(11 196 42 162)(12 195 43 161)(13 194 44 160)(14 193 45 159)(15 192 46 158)(16 191 47 157)(17 190 48 156)(18 189 49 155)(19 206 61 172)(20 205 62 171)(21 204 63 170)(22 203 64 169)(23 202 65 168)(24 201 66 167)(25 200 67 166)(26 199 68 165)(27 216 69 164)(28 215 70 163)(29 214 71 180)(30 213 72 179)(31 212 55 178)(32 211 56 177)(33 210 57 176)(34 209 58 175)(35 208 59 174)(36 207 60 173)(73 257 111 218)(74 256 112 217)(75 255 113 234)(76 254 114 233)(77 253 115 232)(78 270 116 231)(79 269 117 230)(80 268 118 229)(81 267 119 228)(82 266 120 227)(83 265 121 226)(84 264 122 225)(85 263 123 224)(86 262 124 223)(87 261 125 222)(88 260 126 221)(89 259 109 220)(90 258 110 219)(91 285 134 244)(92 284 135 243)(93 283 136 242)(94 282 137 241)(95 281 138 240)(96 280 139 239)(97 279 140 238)(98 278 141 237)(99 277 142 236)(100 276 143 235)(101 275 144 252)(102 274 127 251)(103 273 128 250)(104 272 129 249)(105 271 130 248)(106 288 131 247)(107 287 132 246)(108 286 133 245)
(1 103 31 77)(2 104 32 78)(3 105 33 79)(4 106 34 80)(5 107 35 81)(6 108 36 82)(7 91 19 83)(8 92 20 84)(9 93 21 85)(10 94 22 86)(11 95 23 87)(12 96 24 88)(13 97 25 89)(14 98 26 90)(15 99 27 73)(16 100 28 74)(17 101 29 75)(18 102 30 76)(37 133 60 120)(38 134 61 121)(39 135 62 122)(40 136 63 123)(41 137 64 124)(42 138 65 125)(43 139 66 126)(44 140 67 109)(45 141 68 110)(46 142 69 111)(47 143 70 112)(48 144 71 113)(49 127 72 114)(50 128 55 115)(51 129 56 116)(52 130 57 117)(53 131 58 118)(54 132 59 119)(145 250 169 232)(146 251 170 233)(147 252 171 234)(148 235 172 217)(149 236 173 218)(150 237 174 219)(151 238 175 220)(152 239 176 221)(153 240 177 222)(154 241 178 223)(155 242 179 224)(156 243 180 225)(157 244 163 226)(158 245 164 227)(159 246 165 228)(160 247 166 229)(161 248 167 230)(162 249 168 231)(181 275 205 255)(182 276 206 256)(183 277 207 257)(184 278 208 258)(185 279 209 259)(186 280 210 260)(187 281 211 261)(188 282 212 262)(189 283 213 263)(190 284 214 264)(191 285 215 265)(192 286 216 266)(193 287 199 267)(194 288 200 268)(195 271 201 269)(196 272 202 270)(197 273 203 253)(198 274 204 254)

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

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

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

84 conjugacy classes

class 1 2A···2G 3 4A4B4C4D4E···4L6A···6G9A9B9C12A···12H18A···18U36A···36X
order12···2344444···46···699912···1218···1836···36
size11···12222218···182···22222···22···22···2

84 irreducible representations

dim111112222222222222222
type+++++--++-+-+-+
imageC1C2C2C4C4S3D4Q8Dic3D6D9Dic6C4×S3D12C3⋊D4Dic9D18Dic18C4×D9D36C9⋊D4
kernelC18.C42C22×Dic9C22×C36C2×Dic9C2×C36C22×C12C2×C18C2×C18C2×C12C22×C6C22×C4C2×C6C2×C6C2×C6C2×C6C2×C4C23C22C22C22C22
# reps12184131213242463612612

Matrix representation of C18.C42 in GL4(𝔽37) generated by

1000
0100
00176
003111
,
6000
0100
002533
00812
,
6000
03100
00510
002732
G:=sub<GL(4,GF(37))| [1,0,0,0,0,1,0,0,0,0,17,31,0,0,6,11],[6,0,0,0,0,1,0,0,0,0,25,8,0,0,33,12],[6,0,0,0,0,31,0,0,0,0,5,27,0,0,10,32] >;

C18.C42 in GAP, Magma, Sage, TeX

C_{18}.C_4^2
% in TeX

G:=Group("C18.C4^2");
// GroupNames label

G:=SmallGroup(288,38);
// by ID

G=gap.SmallGroup(288,38);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,253,64,6725,292,9414]);
// Polycyclic

G:=Group<a,b,c|a^18=b^4=c^4=1,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^9*b>;
// generators/relations

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