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G = C4⋊C4×C18order 288 = 25·32

Direct product of C18 and C4⋊C4

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Aliases: C4⋊C4×C18, C42(C2×C36), (C2×C4)⋊3C36, C369(C2×C4), (C2×C36)⋊8C4, C2.2(D4×C18), C6.65(C6×D4), (C2×C18).8Q8, C6.18(C6×Q8), C2.1(Q8×C18), C18.65(C2×D4), (C2×C18).51D4, C18.18(C2×Q8), C12.52(C2×C12), (C2×C12).20C12, C22.3(Q8×C9), (C22×C36).5C2, C2.2(C22×C36), (C22×C4).5C18, C22.13(D4×C9), C23.16(C2×C18), (C22×C12).12C6, C22.11(C2×C36), C6.30(C22×C12), C18.30(C22×C4), (C2×C18).71C23, (C2×C36).119C22, C22.4(C22×C18), (C22×C18).49C22, C3.(C6×C4⋊C4), (C6×C4⋊C4).C3, C6.15(C3×C4⋊C4), (C3×C4⋊C4).23C6, (C2×C4).7(C2×C18), (C2×C6).60(C3×D4), (C2×C6).11(C3×Q8), (C2×C12).80(C2×C6), (C2×C18).40(C2×C4), (C2×C6).49(C2×C12), (C22×C6).74(C2×C6), (C2×C6).76(C22×C6), SmallGroup(288,166)

Series: Derived Chief Lower central Upper central

C1C2 — C4⋊C4×C18
C1C3C6C2×C6C2×C18C2×C36C9×C4⋊C4 — C4⋊C4×C18
C1C2 — C4⋊C4×C18
C1C22×C18 — C4⋊C4×C18

Generators and relations for C4⋊C4×C18
 G = < a,b,c | a18=b4=c4=1, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 162 in 138 conjugacy classes, 114 normal (24 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×4], C4 [×4], C22, C22 [×6], C6 [×3], C6 [×4], C2×C4 [×10], C2×C4 [×4], C23, C9, C12 [×4], C12 [×4], C2×C6, C2×C6 [×6], C4⋊C4 [×4], C22×C4, C22×C4 [×2], C18 [×3], C18 [×4], C2×C12 [×10], C2×C12 [×4], C22×C6, C2×C4⋊C4, C36 [×4], C36 [×4], C2×C18, C2×C18 [×6], C3×C4⋊C4 [×4], C22×C12, C22×C12 [×2], C2×C36 [×10], C2×C36 [×4], C22×C18, C6×C4⋊C4, C9×C4⋊C4 [×4], C22×C36, C22×C36 [×2], C4⋊C4×C18
Quotients: C1, C2 [×7], C3, C4 [×4], C22 [×7], C6 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, C9, C12 [×4], C2×C6 [×7], C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, C18 [×7], C2×C12 [×6], C3×D4 [×2], C3×Q8 [×2], C22×C6, C2×C4⋊C4, C36 [×4], C2×C18 [×7], C3×C4⋊C4 [×4], C22×C12, C6×D4, C6×Q8, C2×C36 [×6], D4×C9 [×2], Q8×C9 [×2], C22×C18, C6×C4⋊C4, C9×C4⋊C4 [×4], C22×C36, D4×C18, Q8×C18, C4⋊C4×C18

Smallest permutation representation of C4⋊C4×C18
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 28 196 64)(2 29 197 65)(3 30 198 66)(4 31 181 67)(5 32 182 68)(6 33 183 69)(7 34 184 70)(8 35 185 71)(9 36 186 72)(10 19 187 55)(11 20 188 56)(12 21 189 57)(13 22 190 58)(14 23 191 59)(15 24 192 60)(16 25 193 61)(17 26 194 62)(18 27 195 63)(37 99 251 254)(38 100 252 255)(39 101 235 256)(40 102 236 257)(41 103 237 258)(42 104 238 259)(43 105 239 260)(44 106 240 261)(45 107 241 262)(46 108 242 263)(47 91 243 264)(48 92 244 265)(49 93 245 266)(50 94 246 267)(51 95 247 268)(52 96 248 269)(53 97 249 270)(54 98 250 253)(73 275 176 162)(74 276 177 145)(75 277 178 146)(76 278 179 147)(77 279 180 148)(78 280 163 149)(79 281 164 150)(80 282 165 151)(81 283 166 152)(82 284 167 153)(83 285 168 154)(84 286 169 155)(85 287 170 156)(86 288 171 157)(87 271 172 158)(88 272 173 159)(89 273 174 160)(90 274 175 161)(109 141 201 221)(110 142 202 222)(111 143 203 223)(112 144 204 224)(113 127 205 225)(114 128 206 226)(115 129 207 227)(116 130 208 228)(117 131 209 229)(118 132 210 230)(119 133 211 231)(120 134 212 232)(121 135 213 233)(122 136 214 234)(123 137 215 217)(124 138 216 218)(125 139 199 219)(126 140 200 220)
(1 165 124 258)(2 166 125 259)(3 167 126 260)(4 168 109 261)(5 169 110 262)(6 170 111 263)(7 171 112 264)(8 172 113 265)(9 173 114 266)(10 174 115 267)(11 175 116 268)(12 176 117 269)(13 177 118 270)(14 178 119 253)(15 179 120 254)(16 180 121 255)(17 163 122 256)(18 164 123 257)(19 273 129 246)(20 274 130 247)(21 275 131 248)(22 276 132 249)(23 277 133 250)(24 278 134 251)(25 279 135 252)(26 280 136 235)(27 281 137 236)(28 282 138 237)(29 283 139 238)(30 284 140 239)(31 285 141 240)(32 286 142 241)(33 287 143 242)(34 288 144 243)(35 271 127 244)(36 272 128 245)(37 60 147 232)(38 61 148 233)(39 62 149 234)(40 63 150 217)(41 64 151 218)(42 65 152 219)(43 66 153 220)(44 67 154 221)(45 68 155 222)(46 69 156 223)(47 70 157 224)(48 71 158 225)(49 72 159 226)(50 55 160 227)(51 56 161 228)(52 57 162 229)(53 58 145 230)(54 59 146 231)(73 209 96 189)(74 210 97 190)(75 211 98 191)(76 212 99 192)(77 213 100 193)(78 214 101 194)(79 215 102 195)(80 216 103 196)(81 199 104 197)(82 200 105 198)(83 201 106 181)(84 202 107 182)(85 203 108 183)(86 204 91 184)(87 205 92 185)(88 206 93 186)(89 207 94 187)(90 208 95 188)

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

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

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

180 conjugacy classes

class 1 2A···2G3A3B4A···4L6A···6N9A···9F12A···12X18A···18AP36A···36BT
order12···2334···46···69···912···1218···1836···36
size11···1112···21···11···12···21···12···2

180 irreducible representations

dim111111111111222222
type++++-
imageC1C2C2C3C4C6C6C9C12C18C18C36D4Q8C3×D4C3×Q8D4×C9Q8×C9
kernelC4⋊C4×C18C9×C4⋊C4C22×C36C6×C4⋊C4C2×C36C3×C4⋊C4C22×C12C2×C4⋊C4C2×C12C4⋊C4C22×C4C2×C4C2×C18C2×C18C2×C6C2×C6C22C22
# reps143288661624184822441212

Matrix representation of C4⋊C4×C18 in GL4(𝔽37) generated by

1000
03600
00330
00033
,
36000
03600
002314
001514
,
6000
0100
002733
001610
G:=sub<GL(4,GF(37))| [1,0,0,0,0,36,0,0,0,0,33,0,0,0,0,33],[36,0,0,0,0,36,0,0,0,0,23,15,0,0,14,14],[6,0,0,0,0,1,0,0,0,0,27,16,0,0,33,10] >;

C4⋊C4×C18 in GAP, Magma, Sage, TeX

C_4\rtimes C_4\times C_{18}
% in TeX

G:=Group("C4:C4xC18");
// GroupNames label

G:=SmallGroup(288,166);
// by ID

G=gap.SmallGroup(288,166);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,336,365,176,360]);
// Polycyclic

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

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