direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C4⋊C4×C18, C4⋊2(C2×C36), (C2×C4)⋊3C36, C36⋊9(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
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, C2, C3, C4, C4, C22, C22, C6, C6, C2×C4, C2×C4, C23, C9, C12, C12, C2×C6, C2×C6, C4⋊C4, C22×C4, C22×C4, C18, C18, C2×C12, C2×C12, C22×C6, C2×C4⋊C4, C36, C36, C2×C18, C2×C18, C3×C4⋊C4, C22×C12, C22×C12, C2×C36, C2×C36, C22×C18, C6×C4⋊C4, C9×C4⋊C4, C22×C36, C22×C36, C4⋊C4×C18
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, Q8, C23, C9, C12, C2×C6, C4⋊C4, C22×C4, C2×D4, C2×Q8, C18, C2×C12, C3×D4, C3×Q8, C22×C6, C2×C4⋊C4, C36, C2×C18, C3×C4⋊C4, C22×C12, C6×D4, C6×Q8, C2×C36, D4×C9, Q8×C9, C22×C18, C6×C4⋊C4, C9×C4⋊C4, C22×C36, D4×C18, Q8×C18, 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)
(1 280 80 223)(2 281 81 224)(3 282 82 225)(4 283 83 226)(5 284 84 227)(6 285 85 228)(7 286 86 229)(8 287 87 230)(9 288 88 231)(10 271 89 232)(11 272 90 233)(12 273 73 234)(13 274 74 217)(14 275 75 218)(15 276 76 219)(16 277 77 220)(17 278 78 221)(18 279 79 222)(19 200 254 101)(20 201 255 102)(21 202 256 103)(22 203 257 104)(23 204 258 105)(24 205 259 106)(25 206 260 107)(26 207 261 108)(27 208 262 91)(28 209 263 92)(29 210 264 93)(30 211 265 94)(31 212 266 95)(32 213 267 96)(33 214 268 97)(34 215 269 98)(35 216 270 99)(36 199 253 100)(37 164 188 241)(38 165 189 242)(39 166 190 243)(40 167 191 244)(41 168 192 245)(42 169 193 246)(43 170 194 247)(44 171 195 248)(45 172 196 249)(46 173 197 250)(47 174 198 251)(48 175 181 252)(49 176 182 235)(50 177 183 236)(51 178 184 237)(52 179 185 238)(53 180 186 239)(54 163 187 240)(55 134 126 149)(56 135 109 150)(57 136 110 151)(58 137 111 152)(59 138 112 153)(60 139 113 154)(61 140 114 155)(62 141 115 156)(63 142 116 157)(64 143 117 158)(65 144 118 159)(66 127 119 160)(67 128 120 161)(68 129 121 162)(69 130 122 145)(70 131 123 146)(71 132 124 147)(72 133 125 148)
(1 182 265 122)(2 183 266 123)(3 184 267 124)(4 185 268 125)(5 186 269 126)(6 187 270 109)(7 188 253 110)(8 189 254 111)(9 190 255 112)(10 191 256 113)(11 192 257 114)(12 193 258 115)(13 194 259 116)(14 195 260 117)(15 196 261 118)(16 197 262 119)(17 198 263 120)(18 181 264 121)(19 58 87 38)(20 59 88 39)(21 60 89 40)(22 61 90 41)(23 62 73 42)(24 63 74 43)(25 64 75 44)(26 65 76 45)(27 66 77 46)(28 67 78 47)(29 68 79 48)(30 69 80 49)(31 70 81 50)(32 71 82 51)(33 72 83 52)(34 55 84 53)(35 56 85 54)(36 57 86 37)(91 127 277 173)(92 128 278 174)(93 129 279 175)(94 130 280 176)(95 131 281 177)(96 132 282 178)(97 133 283 179)(98 134 284 180)(99 135 285 163)(100 136 286 164)(101 137 287 165)(102 138 288 166)(103 139 271 167)(104 140 272 168)(105 141 273 169)(106 142 274 170)(107 143 275 171)(108 144 276 172)(145 223 235 211)(146 224 236 212)(147 225 237 213)(148 226 238 214)(149 227 239 215)(150 228 240 216)(151 229 241 199)(152 230 242 200)(153 231 243 201)(154 232 244 202)(155 233 245 203)(156 234 246 204)(157 217 247 205)(158 218 248 206)(159 219 249 207)(160 220 250 208)(161 221 251 209)(162 222 252 210)
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), (1,280,80,223)(2,281,81,224)(3,282,82,225)(4,283,83,226)(5,284,84,227)(6,285,85,228)(7,286,86,229)(8,287,87,230)(9,288,88,231)(10,271,89,232)(11,272,90,233)(12,273,73,234)(13,274,74,217)(14,275,75,218)(15,276,76,219)(16,277,77,220)(17,278,78,221)(18,279,79,222)(19,200,254,101)(20,201,255,102)(21,202,256,103)(22,203,257,104)(23,204,258,105)(24,205,259,106)(25,206,260,107)(26,207,261,108)(27,208,262,91)(28,209,263,92)(29,210,264,93)(30,211,265,94)(31,212,266,95)(32,213,267,96)(33,214,268,97)(34,215,269,98)(35,216,270,99)(36,199,253,100)(37,164,188,241)(38,165,189,242)(39,166,190,243)(40,167,191,244)(41,168,192,245)(42,169,193,246)(43,170,194,247)(44,171,195,248)(45,172,196,249)(46,173,197,250)(47,174,198,251)(48,175,181,252)(49,176,182,235)(50,177,183,236)(51,178,184,237)(52,179,185,238)(53,180,186,239)(54,163,187,240)(55,134,126,149)(56,135,109,150)(57,136,110,151)(58,137,111,152)(59,138,112,153)(60,139,113,154)(61,140,114,155)(62,141,115,156)(63,142,116,157)(64,143,117,158)(65,144,118,159)(66,127,119,160)(67,128,120,161)(68,129,121,162)(69,130,122,145)(70,131,123,146)(71,132,124,147)(72,133,125,148), (1,182,265,122)(2,183,266,123)(3,184,267,124)(4,185,268,125)(5,186,269,126)(6,187,270,109)(7,188,253,110)(8,189,254,111)(9,190,255,112)(10,191,256,113)(11,192,257,114)(12,193,258,115)(13,194,259,116)(14,195,260,117)(15,196,261,118)(16,197,262,119)(17,198,263,120)(18,181,264,121)(19,58,87,38)(20,59,88,39)(21,60,89,40)(22,61,90,41)(23,62,73,42)(24,63,74,43)(25,64,75,44)(26,65,76,45)(27,66,77,46)(28,67,78,47)(29,68,79,48)(30,69,80,49)(31,70,81,50)(32,71,82,51)(33,72,83,52)(34,55,84,53)(35,56,85,54)(36,57,86,37)(91,127,277,173)(92,128,278,174)(93,129,279,175)(94,130,280,176)(95,131,281,177)(96,132,282,178)(97,133,283,179)(98,134,284,180)(99,135,285,163)(100,136,286,164)(101,137,287,165)(102,138,288,166)(103,139,271,167)(104,140,272,168)(105,141,273,169)(106,142,274,170)(107,143,275,171)(108,144,276,172)(145,223,235,211)(146,224,236,212)(147,225,237,213)(148,226,238,214)(149,227,239,215)(150,228,240,216)(151,229,241,199)(152,230,242,200)(153,231,243,201)(154,232,244,202)(155,233,245,203)(156,234,246,204)(157,217,247,205)(158,218,248,206)(159,219,249,207)(160,220,250,208)(161,221,251,209)(162,222,252,210)>;
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), (1,280,80,223)(2,281,81,224)(3,282,82,225)(4,283,83,226)(5,284,84,227)(6,285,85,228)(7,286,86,229)(8,287,87,230)(9,288,88,231)(10,271,89,232)(11,272,90,233)(12,273,73,234)(13,274,74,217)(14,275,75,218)(15,276,76,219)(16,277,77,220)(17,278,78,221)(18,279,79,222)(19,200,254,101)(20,201,255,102)(21,202,256,103)(22,203,257,104)(23,204,258,105)(24,205,259,106)(25,206,260,107)(26,207,261,108)(27,208,262,91)(28,209,263,92)(29,210,264,93)(30,211,265,94)(31,212,266,95)(32,213,267,96)(33,214,268,97)(34,215,269,98)(35,216,270,99)(36,199,253,100)(37,164,188,241)(38,165,189,242)(39,166,190,243)(40,167,191,244)(41,168,192,245)(42,169,193,246)(43,170,194,247)(44,171,195,248)(45,172,196,249)(46,173,197,250)(47,174,198,251)(48,175,181,252)(49,176,182,235)(50,177,183,236)(51,178,184,237)(52,179,185,238)(53,180,186,239)(54,163,187,240)(55,134,126,149)(56,135,109,150)(57,136,110,151)(58,137,111,152)(59,138,112,153)(60,139,113,154)(61,140,114,155)(62,141,115,156)(63,142,116,157)(64,143,117,158)(65,144,118,159)(66,127,119,160)(67,128,120,161)(68,129,121,162)(69,130,122,145)(70,131,123,146)(71,132,124,147)(72,133,125,148), (1,182,265,122)(2,183,266,123)(3,184,267,124)(4,185,268,125)(5,186,269,126)(6,187,270,109)(7,188,253,110)(8,189,254,111)(9,190,255,112)(10,191,256,113)(11,192,257,114)(12,193,258,115)(13,194,259,116)(14,195,260,117)(15,196,261,118)(16,197,262,119)(17,198,263,120)(18,181,264,121)(19,58,87,38)(20,59,88,39)(21,60,89,40)(22,61,90,41)(23,62,73,42)(24,63,74,43)(25,64,75,44)(26,65,76,45)(27,66,77,46)(28,67,78,47)(29,68,79,48)(30,69,80,49)(31,70,81,50)(32,71,82,51)(33,72,83,52)(34,55,84,53)(35,56,85,54)(36,57,86,37)(91,127,277,173)(92,128,278,174)(93,129,279,175)(94,130,280,176)(95,131,281,177)(96,132,282,178)(97,133,283,179)(98,134,284,180)(99,135,285,163)(100,136,286,164)(101,137,287,165)(102,138,288,166)(103,139,271,167)(104,140,272,168)(105,141,273,169)(106,142,274,170)(107,143,275,171)(108,144,276,172)(145,223,235,211)(146,224,236,212)(147,225,237,213)(148,226,238,214)(149,227,239,215)(150,228,240,216)(151,229,241,199)(152,230,242,200)(153,231,243,201)(154,232,244,202)(155,233,245,203)(156,234,246,204)(157,217,247,205)(158,218,248,206)(159,219,249,207)(160,220,250,208)(161,221,251,209)(162,222,252,210) );
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)], [(1,280,80,223),(2,281,81,224),(3,282,82,225),(4,283,83,226),(5,284,84,227),(6,285,85,228),(7,286,86,229),(8,287,87,230),(9,288,88,231),(10,271,89,232),(11,272,90,233),(12,273,73,234),(13,274,74,217),(14,275,75,218),(15,276,76,219),(16,277,77,220),(17,278,78,221),(18,279,79,222),(19,200,254,101),(20,201,255,102),(21,202,256,103),(22,203,257,104),(23,204,258,105),(24,205,259,106),(25,206,260,107),(26,207,261,108),(27,208,262,91),(28,209,263,92),(29,210,264,93),(30,211,265,94),(31,212,266,95),(32,213,267,96),(33,214,268,97),(34,215,269,98),(35,216,270,99),(36,199,253,100),(37,164,188,241),(38,165,189,242),(39,166,190,243),(40,167,191,244),(41,168,192,245),(42,169,193,246),(43,170,194,247),(44,171,195,248),(45,172,196,249),(46,173,197,250),(47,174,198,251),(48,175,181,252),(49,176,182,235),(50,177,183,236),(51,178,184,237),(52,179,185,238),(53,180,186,239),(54,163,187,240),(55,134,126,149),(56,135,109,150),(57,136,110,151),(58,137,111,152),(59,138,112,153),(60,139,113,154),(61,140,114,155),(62,141,115,156),(63,142,116,157),(64,143,117,158),(65,144,118,159),(66,127,119,160),(67,128,120,161),(68,129,121,162),(69,130,122,145),(70,131,123,146),(71,132,124,147),(72,133,125,148)], [(1,182,265,122),(2,183,266,123),(3,184,267,124),(4,185,268,125),(5,186,269,126),(6,187,270,109),(7,188,253,110),(8,189,254,111),(9,190,255,112),(10,191,256,113),(11,192,257,114),(12,193,258,115),(13,194,259,116),(14,195,260,117),(15,196,261,118),(16,197,262,119),(17,198,263,120),(18,181,264,121),(19,58,87,38),(20,59,88,39),(21,60,89,40),(22,61,90,41),(23,62,73,42),(24,63,74,43),(25,64,75,44),(26,65,76,45),(27,66,77,46),(28,67,78,47),(29,68,79,48),(30,69,80,49),(31,70,81,50),(32,71,82,51),(33,72,83,52),(34,55,84,53),(35,56,85,54),(36,57,86,37),(91,127,277,173),(92,128,278,174),(93,129,279,175),(94,130,280,176),(95,131,281,177),(96,132,282,178),(97,133,283,179),(98,134,284,180),(99,135,285,163),(100,136,286,164),(101,137,287,165),(102,138,288,166),(103,139,271,167),(104,140,272,168),(105,141,273,169),(106,142,274,170),(107,143,275,171),(108,144,276,172),(145,223,235,211),(146,224,236,212),(147,225,237,213),(148,226,238,214),(149,227,239,215),(150,228,240,216),(151,229,241,199),(152,230,242,200),(153,231,243,201),(154,232,244,202),(155,233,245,203),(156,234,246,204),(157,217,247,205),(158,218,248,206),(159,219,249,207),(160,220,250,208),(161,221,251,209),(162,222,252,210)]])
180 conjugacy classes
| class | 1 | 2A | ··· | 2G | 3A | 3B | 4A | ··· | 4L | 6A | ··· | 6N | 9A | ··· | 9F | 12A | ··· | 12X | 18A | ··· | 18AP | 36A | ··· | 36BT |
| order | 1 | 2 | ··· | 2 | 3 | 3 | 4 | ··· | 4 | 6 | ··· | 6 | 9 | ··· | 9 | 12 | ··· | 12 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | ··· | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
180 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | |||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C6 | C6 | C9 | C12 | C18 | C18 | C36 | D4 | Q8 | C3×D4 | C3×Q8 | D4×C9 | Q8×C9 |
| kernel | C4⋊C4×C18 | C9×C4⋊C4 | C22×C36 | C6×C4⋊C4 | C2×C36 | C3×C4⋊C4 | C22×C12 | C2×C4⋊C4 | C2×C12 | C4⋊C4 | C22×C4 | C2×C4 | C2×C18 | C2×C18 | C2×C6 | C2×C6 | C22 | C22 |
| # reps | 1 | 4 | 3 | 2 | 8 | 8 | 6 | 6 | 16 | 24 | 18 | 48 | 2 | 2 | 4 | 4 | 12 | 12 |
Matrix representation of C4⋊C4×C18 ►in GL4(𝔽37) generated by
| 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 | 14 |
| 0 | 0 | 15 | 14 |
| 6 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 27 | 33 |
| 0 | 0 | 16 | 10 |
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