metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C36.2Q8, C12.2Dic6, C4.2Dic18, C18.4SD16, C9⋊C8⋊2C4, C4⋊C4.2D9, C9⋊1(C4.Q8), C12.2(C4×S3), C36.2(C2×C4), C4.12(C4×D9), C18.3(C4⋊C4), (C2×C12).38D6, (C2×C18).30D4, (C2×C4).36D18, C4⋊Dic9.8C2, C6.7(D4.S3), C2.1(D4.D9), C3.(C12.Q8), C2.4(Dic9⋊C4), (C2×C36).16C22, C6.7(Q8⋊2S3), C2.1(Q8⋊2D9), C6.11(Dic3⋊C4), C22.13(C9⋊D4), (C2×C9⋊C8).2C2, (C9×C4⋊C4).2C2, (C3×C4⋊C4).2S3, (C2×C6).68(C3⋊D4), SmallGroup(288,15)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4.Dic18
G = < a,b,c | a4=b36=1, c2=ab18, bab-1=a-1, ac=ca, cbc-1=a-1b-1 >
Smallest permutation representation of C4.Dic18
►Regular action on 288 points
Generators in S288
(1 126 158 45)(2 46 159 127)(3 128 160 47)(4 48 161 129)(5 130 162 49)(6 50 163 131)(7 132 164 51)(8 52 165 133)(9 134 166 53)(10 54 167 135)(11 136 168 55)(12 56 169 137)(13 138 170 57)(14 58 171 139)(15 140 172 59)(16 60 173 141)(17 142 174 61)(18 62 175 143)(19 144 176 63)(20 64 177 109)(21 110 178 65)(22 66 179 111)(23 112 180 67)(24 68 145 113)(25 114 146 69)(26 70 147 115)(27 116 148 71)(28 72 149 117)(29 118 150 37)(30 38 151 119)(31 120 152 39)(32 40 153 121)(33 122 154 41)(34 42 155 123)(35 124 156 43)(36 44 157 125)(73 194 267 245)(74 246 268 195)(75 196 269 247)(76 248 270 197)(77 198 271 249)(78 250 272 199)(79 200 273 251)(80 252 274 201)(81 202 275 217)(82 218 276 203)(83 204 277 219)(84 220 278 205)(85 206 279 221)(86 222 280 207)(87 208 281 223)(88 224 282 209)(89 210 283 225)(90 226 284 211)(91 212 285 227)(92 228 286 213)(93 214 287 229)(94 230 288 215)(95 216 253 231)(96 232 254 181)(97 182 255 233)(98 234 256 183)(99 184 257 235)(100 236 258 185)(101 186 259 237)(102 238 260 187)(103 188 261 239)(104 240 262 189)(105 190 263 241)(106 242 264 191)(107 192 265 243)(108 244 266 193)
(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)
(1 212 144 267 158 227 63 73)(2 284 64 193 159 90 109 244)(3 210 110 265 160 225 65 107)(4 282 66 191 161 88 111 242)(5 208 112 263 162 223 67 105)(6 280 68 189 163 86 113 240)(7 206 114 261 164 221 69 103)(8 278 70 187 165 84 115 238)(9 204 116 259 166 219 71 101)(10 276 72 185 167 82 117 236)(11 202 118 257 168 217 37 99)(12 274 38 183 169 80 119 234)(13 200 120 255 170 251 39 97)(14 272 40 181 171 78 121 232)(15 198 122 253 172 249 41 95)(16 270 42 215 173 76 123 230)(17 196 124 287 174 247 43 93)(18 268 44 213 175 74 125 228)(19 194 126 285 176 245 45 91)(20 266 46 211 177 108 127 226)(21 192 128 283 178 243 47 89)(22 264 48 209 179 106 129 224)(23 190 130 281 180 241 49 87)(24 262 50 207 145 104 131 222)(25 188 132 279 146 239 51 85)(26 260 52 205 147 102 133 220)(27 186 134 277 148 237 53 83)(28 258 54 203 149 100 135 218)(29 184 136 275 150 235 55 81)(30 256 56 201 151 98 137 252)(31 182 138 273 152 233 57 79)(32 254 58 199 153 96 139 250)(33 216 140 271 154 231 59 77)(34 288 60 197 155 94 141 248)(35 214 142 269 156 229 61 75)(36 286 62 195 157 92 143 246)
G:=sub<Sym(288)| (1,126,158,45)(2,46,159,127)(3,128,160,47)(4,48,161,129)(5,130,162,49)(6,50,163,131)(7,132,164,51)(8,52,165,133)(9,134,166,53)(10,54,167,135)(11,136,168,55)(12,56,169,137)(13,138,170,57)(14,58,171,139)(15,140,172,59)(16,60,173,141)(17,142,174,61)(18,62,175,143)(19,144,176,63)(20,64,177,109)(21,110,178,65)(22,66,179,111)(23,112,180,67)(24,68,145,113)(25,114,146,69)(26,70,147,115)(27,116,148,71)(28,72,149,117)(29,118,150,37)(30,38,151,119)(31,120,152,39)(32,40,153,121)(33,122,154,41)(34,42,155,123)(35,124,156,43)(36,44,157,125)(73,194,267,245)(74,246,268,195)(75,196,269,247)(76,248,270,197)(77,198,271,249)(78,250,272,199)(79,200,273,251)(80,252,274,201)(81,202,275,217)(82,218,276,203)(83,204,277,219)(84,220,278,205)(85,206,279,221)(86,222,280,207)(87,208,281,223)(88,224,282,209)(89,210,283,225)(90,226,284,211)(91,212,285,227)(92,228,286,213)(93,214,287,229)(94,230,288,215)(95,216,253,231)(96,232,254,181)(97,182,255,233)(98,234,256,183)(99,184,257,235)(100,236,258,185)(101,186,259,237)(102,238,260,187)(103,188,261,239)(104,240,262,189)(105,190,263,241)(106,242,264,191)(107,192,265,243)(108,244,266,193), (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), (1,212,144,267,158,227,63,73)(2,284,64,193,159,90,109,244)(3,210,110,265,160,225,65,107)(4,282,66,191,161,88,111,242)(5,208,112,263,162,223,67,105)(6,280,68,189,163,86,113,240)(7,206,114,261,164,221,69,103)(8,278,70,187,165,84,115,238)(9,204,116,259,166,219,71,101)(10,276,72,185,167,82,117,236)(11,202,118,257,168,217,37,99)(12,274,38,183,169,80,119,234)(13,200,120,255,170,251,39,97)(14,272,40,181,171,78,121,232)(15,198,122,253,172,249,41,95)(16,270,42,215,173,76,123,230)(17,196,124,287,174,247,43,93)(18,268,44,213,175,74,125,228)(19,194,126,285,176,245,45,91)(20,266,46,211,177,108,127,226)(21,192,128,283,178,243,47,89)(22,264,48,209,179,106,129,224)(23,190,130,281,180,241,49,87)(24,262,50,207,145,104,131,222)(25,188,132,279,146,239,51,85)(26,260,52,205,147,102,133,220)(27,186,134,277,148,237,53,83)(28,258,54,203,149,100,135,218)(29,184,136,275,150,235,55,81)(30,256,56,201,151,98,137,252)(31,182,138,273,152,233,57,79)(32,254,58,199,153,96,139,250)(33,216,140,271,154,231,59,77)(34,288,60,197,155,94,141,248)(35,214,142,269,156,229,61,75)(36,286,62,195,157,92,143,246)>;
G:=Group( (1,126,158,45)(2,46,159,127)(3,128,160,47)(4,48,161,129)(5,130,162,49)(6,50,163,131)(7,132,164,51)(8,52,165,133)(9,134,166,53)(10,54,167,135)(11,136,168,55)(12,56,169,137)(13,138,170,57)(14,58,171,139)(15,140,172,59)(16,60,173,141)(17,142,174,61)(18,62,175,143)(19,144,176,63)(20,64,177,109)(21,110,178,65)(22,66,179,111)(23,112,180,67)(24,68,145,113)(25,114,146,69)(26,70,147,115)(27,116,148,71)(28,72,149,117)(29,118,150,37)(30,38,151,119)(31,120,152,39)(32,40,153,121)(33,122,154,41)(34,42,155,123)(35,124,156,43)(36,44,157,125)(73,194,267,245)(74,246,268,195)(75,196,269,247)(76,248,270,197)(77,198,271,249)(78,250,272,199)(79,200,273,251)(80,252,274,201)(81,202,275,217)(82,218,276,203)(83,204,277,219)(84,220,278,205)(85,206,279,221)(86,222,280,207)(87,208,281,223)(88,224,282,209)(89,210,283,225)(90,226,284,211)(91,212,285,227)(92,228,286,213)(93,214,287,229)(94,230,288,215)(95,216,253,231)(96,232,254,181)(97,182,255,233)(98,234,256,183)(99,184,257,235)(100,236,258,185)(101,186,259,237)(102,238,260,187)(103,188,261,239)(104,240,262,189)(105,190,263,241)(106,242,264,191)(107,192,265,243)(108,244,266,193), (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), (1,212,144,267,158,227,63,73)(2,284,64,193,159,90,109,244)(3,210,110,265,160,225,65,107)(4,282,66,191,161,88,111,242)(5,208,112,263,162,223,67,105)(6,280,68,189,163,86,113,240)(7,206,114,261,164,221,69,103)(8,278,70,187,165,84,115,238)(9,204,116,259,166,219,71,101)(10,276,72,185,167,82,117,236)(11,202,118,257,168,217,37,99)(12,274,38,183,169,80,119,234)(13,200,120,255,170,251,39,97)(14,272,40,181,171,78,121,232)(15,198,122,253,172,249,41,95)(16,270,42,215,173,76,123,230)(17,196,124,287,174,247,43,93)(18,268,44,213,175,74,125,228)(19,194,126,285,176,245,45,91)(20,266,46,211,177,108,127,226)(21,192,128,283,178,243,47,89)(22,264,48,209,179,106,129,224)(23,190,130,281,180,241,49,87)(24,262,50,207,145,104,131,222)(25,188,132,279,146,239,51,85)(26,260,52,205,147,102,133,220)(27,186,134,277,148,237,53,83)(28,258,54,203,149,100,135,218)(29,184,136,275,150,235,55,81)(30,256,56,201,151,98,137,252)(31,182,138,273,152,233,57,79)(32,254,58,199,153,96,139,250)(33,216,140,271,154,231,59,77)(34,288,60,197,155,94,141,248)(35,214,142,269,156,229,61,75)(36,286,62,195,157,92,143,246) );
G=PermutationGroup([[(1,126,158,45),(2,46,159,127),(3,128,160,47),(4,48,161,129),(5,130,162,49),(6,50,163,131),(7,132,164,51),(8,52,165,133),(9,134,166,53),(10,54,167,135),(11,136,168,55),(12,56,169,137),(13,138,170,57),(14,58,171,139),(15,140,172,59),(16,60,173,141),(17,142,174,61),(18,62,175,143),(19,144,176,63),(20,64,177,109),(21,110,178,65),(22,66,179,111),(23,112,180,67),(24,68,145,113),(25,114,146,69),(26,70,147,115),(27,116,148,71),(28,72,149,117),(29,118,150,37),(30,38,151,119),(31,120,152,39),(32,40,153,121),(33,122,154,41),(34,42,155,123),(35,124,156,43),(36,44,157,125),(73,194,267,245),(74,246,268,195),(75,196,269,247),(76,248,270,197),(77,198,271,249),(78,250,272,199),(79,200,273,251),(80,252,274,201),(81,202,275,217),(82,218,276,203),(83,204,277,219),(84,220,278,205),(85,206,279,221),(86,222,280,207),(87,208,281,223),(88,224,282,209),(89,210,283,225),(90,226,284,211),(91,212,285,227),(92,228,286,213),(93,214,287,229),(94,230,288,215),(95,216,253,231),(96,232,254,181),(97,182,255,233),(98,234,256,183),(99,184,257,235),(100,236,258,185),(101,186,259,237),(102,238,260,187),(103,188,261,239),(104,240,262,189),(105,190,263,241),(106,242,264,191),(107,192,265,243),(108,244,266,193)], [(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)], [(1,212,144,267,158,227,63,73),(2,284,64,193,159,90,109,244),(3,210,110,265,160,225,65,107),(4,282,66,191,161,88,111,242),(5,208,112,263,162,223,67,105),(6,280,68,189,163,86,113,240),(7,206,114,261,164,221,69,103),(8,278,70,187,165,84,115,238),(9,204,116,259,166,219,71,101),(10,276,72,185,167,82,117,236),(11,202,118,257,168,217,37,99),(12,274,38,183,169,80,119,234),(13,200,120,255,170,251,39,97),(14,272,40,181,171,78,121,232),(15,198,122,253,172,249,41,95),(16,270,42,215,173,76,123,230),(17,196,124,287,174,247,43,93),(18,268,44,213,175,74,125,228),(19,194,126,285,176,245,45,91),(20,266,46,211,177,108,127,226),(21,192,128,283,178,243,47,89),(22,264,48,209,179,106,129,224),(23,190,130,281,180,241,49,87),(24,262,50,207,145,104,131,222),(25,188,132,279,146,239,51,85),(26,260,52,205,147,102,133,220),(27,186,134,277,148,237,53,83),(28,258,54,203,149,100,135,218),(29,184,136,275,150,235,55,81),(30,256,56,201,151,98,137,252),(31,182,138,273,152,233,57,79),(32,254,58,199,153,96,139,250),(33,216,140,271,154,231,59,77),(34,288,60,197,155,94,141,248),(35,214,142,269,156,229,61,75),(36,286,62,195,157,92,143,246)]])
54 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 | ··· | 12F | 18A | ··· | 18I | 36A | ··· | 36R |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 9 | 9 | 9 | 12 | ··· | 12 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 36 | 36 | 2 | 2 | 2 | 18 | 18 | 18 | 18 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | - | + | + | + | - | + | - | - | + | - | + | ||||||
| image | C1 | C2 | C2 | C2 | C4 | S3 | Q8 | D4 | D6 | SD16 | D9 | Dic6 | C4×S3 | C3⋊D4 | D18 | Dic18 | C4×D9 | C9⋊D4 | D4.S3 | Q8⋊2S3 | D4.D9 | Q8⋊2D9 |
| kernel | C4.Dic18 | C2×C9⋊C8 | C4⋊Dic9 | C9×C4⋊C4 | C9⋊C8 | C3×C4⋊C4 | C36 | C2×C18 | C2×C12 | C18 | C4⋊C4 | C12 | C12 | C2×C6 | C2×C4 | C4 | C4 | C22 | C6 | C6 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 1 | 1 | 4 | 3 | 2 | 2 | 2 | 3 | 6 | 6 | 6 | 1 | 1 | 3 | 3 |
Matrix representation of C4.Dic18 ►in GL4(𝔽73) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 72 |
| 0 | 0 | 1 | 0 |
| 54 | 25 | 0 | 0 |
| 48 | 29 | 0 | 0 |
| 0 | 0 | 7 | 13 |
| 0 | 0 | 13 | 66 |
| 14 | 19 | 0 | 0 |
| 5 | 59 | 0 | 0 |
| 0 | 0 | 6 | 67 |
| 0 | 0 | 6 | 6 |
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,0,1,0,0,72,0],[54,48,0,0,25,29,0,0,0,0,7,13,0,0,13,66],[14,5,0,0,19,59,0,0,0,0,6,6,0,0,67,6] >;
C4.Dic18 in GAP, Magma, Sage, TeX
C_4.{\rm Dic}_{18} % in TeX
G:=Group("C4.Dic18"); // GroupNames label
G:=SmallGroup(288,15);
// by ID
G=gap.SmallGroup(288,15);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,365,36,346,80,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c|a^4=b^36=1,c^2=a*b^18,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^-1*b^-1>;
// generators/relations
Export