metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C36.45D4, C18.1Q16, Dic18⋊2C4, C2.1Dic36, C18.1SD16, C6.1Dic12, C22.7D36, (C2×C8).2D9, C4.7(C4×D9), (C2×C72).2C2, (C2×C24).2S3, C12.55(C4×S3), C36.17(C2×C4), C9⋊2(Q8⋊C4), (C2×C4).73D18, (C2×C18).12D4, (C2×C6).20D12, C4⋊Dic9.1C2, C6.1(C24⋊C2), C2.7(D18⋊C4), C6.12(D6⋊C4), (C2×C12).362D6, C2.1(C72⋊C2), C4.19(C9⋊D4), C3.(C2.Dic12), C18.5(C22⋊C4), (C2×C36).81C22, (C2×Dic18).1C2, C12.106(C3⋊D4), SmallGroup(288,24)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C36.45D4
G = < a,b,c | a36=b4=1, c2=a18, bab-1=cac-1=a-1, cbc-1=a9b-1 >
Subgroups: 280 in 63 conjugacy classes, 32 normal (30 characteristic)
C1, C2, C3, C4, C4, C22, C6, C8, C2×C4, C2×C4, Q8, C9, Dic3, C12, C2×C6, C4⋊C4, C2×C8, C2×Q8, C18, C24, Dic6, C2×Dic3, C2×C12, Q8⋊C4, Dic9, C36, C2×C18, C4⋊Dic3, C2×C24, C2×Dic6, C72, Dic18, Dic18, C2×Dic9, C2×C36, C2.Dic12, C4⋊Dic9, C2×C72, C2×Dic18, C36.45D4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D6, C22⋊C4, SD16, Q16, D9, C4×S3, D12, C3⋊D4, Q8⋊C4, D18, C24⋊C2, Dic12, D6⋊C4, C4×D9, D36, C9⋊D4, C2.Dic12, Dic36, C72⋊C2, D18⋊C4, C36.45D4
►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 58 266 94)(2 57 267 93)(3 56 268 92)(4 55 269 91)(5 54 270 90)(6 53 271 89)(7 52 272 88)(8 51 273 87)(9 50 274 86)(10 49 275 85)(11 48 276 84)(12 47 277 83)(13 46 278 82)(14 45 279 81)(15 44 280 80)(16 43 281 79)(17 42 282 78)(18 41 283 77)(19 40 284 76)(20 39 285 75)(21 38 286 74)(22 37 287 73)(23 72 288 108)(24 71 253 107)(25 70 254 106)(26 69 255 105)(27 68 256 104)(28 67 257 103)(29 66 258 102)(30 65 259 101)(31 64 260 100)(32 63 261 99)(33 62 262 98)(34 61 263 97)(35 60 264 96)(36 59 265 95)(109 187 217 153)(110 186 218 152)(111 185 219 151)(112 184 220 150)(113 183 221 149)(114 182 222 148)(115 181 223 147)(116 216 224 146)(117 215 225 145)(118 214 226 180)(119 213 227 179)(120 212 228 178)(121 211 229 177)(122 210 230 176)(123 209 231 175)(124 208 232 174)(125 207 233 173)(126 206 234 172)(127 205 235 171)(128 204 236 170)(129 203 237 169)(130 202 238 168)(131 201 239 167)(132 200 240 166)(133 199 241 165)(134 198 242 164)(135 197 243 163)(136 196 244 162)(137 195 245 161)(138 194 246 160)(139 193 247 159)(140 192 248 158)(141 191 249 157)(142 190 250 156)(143 189 251 155)(144 188 252 154)
(1 140 19 122)(2 139 20 121)(3 138 21 120)(4 137 22 119)(5 136 23 118)(6 135 24 117)(7 134 25 116)(8 133 26 115)(9 132 27 114)(10 131 28 113)(11 130 29 112)(12 129 30 111)(13 128 31 110)(14 127 32 109)(15 126 33 144)(16 125 34 143)(17 124 35 142)(18 123 36 141)(37 170 55 152)(38 169 56 151)(39 168 57 150)(40 167 58 149)(41 166 59 148)(42 165 60 147)(43 164 61 146)(44 163 62 145)(45 162 63 180)(46 161 64 179)(47 160 65 178)(48 159 66 177)(49 158 67 176)(50 157 68 175)(51 156 69 174)(52 155 70 173)(53 154 71 172)(54 153 72 171)(73 204 91 186)(74 203 92 185)(75 202 93 184)(76 201 94 183)(77 200 95 182)(78 199 96 181)(79 198 97 216)(80 197 98 215)(81 196 99 214)(82 195 100 213)(83 194 101 212)(84 193 102 211)(85 192 103 210)(86 191 104 209)(87 190 105 208)(88 189 106 207)(89 188 107 206)(90 187 108 205)(217 279 235 261)(218 278 236 260)(219 277 237 259)(220 276 238 258)(221 275 239 257)(222 274 240 256)(223 273 241 255)(224 272 242 254)(225 271 243 253)(226 270 244 288)(227 269 245 287)(228 268 246 286)(229 267 247 285)(230 266 248 284)(231 265 249 283)(232 264 250 282)(233 263 251 281)(234 262 252 280)
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), (1,58,266,94)(2,57,267,93)(3,56,268,92)(4,55,269,91)(5,54,270,90)(6,53,271,89)(7,52,272,88)(8,51,273,87)(9,50,274,86)(10,49,275,85)(11,48,276,84)(12,47,277,83)(13,46,278,82)(14,45,279,81)(15,44,280,80)(16,43,281,79)(17,42,282,78)(18,41,283,77)(19,40,284,76)(20,39,285,75)(21,38,286,74)(22,37,287,73)(23,72,288,108)(24,71,253,107)(25,70,254,106)(26,69,255,105)(27,68,256,104)(28,67,257,103)(29,66,258,102)(30,65,259,101)(31,64,260,100)(32,63,261,99)(33,62,262,98)(34,61,263,97)(35,60,264,96)(36,59,265,95)(109,187,217,153)(110,186,218,152)(111,185,219,151)(112,184,220,150)(113,183,221,149)(114,182,222,148)(115,181,223,147)(116,216,224,146)(117,215,225,145)(118,214,226,180)(119,213,227,179)(120,212,228,178)(121,211,229,177)(122,210,230,176)(123,209,231,175)(124,208,232,174)(125,207,233,173)(126,206,234,172)(127,205,235,171)(128,204,236,170)(129,203,237,169)(130,202,238,168)(131,201,239,167)(132,200,240,166)(133,199,241,165)(134,198,242,164)(135,197,243,163)(136,196,244,162)(137,195,245,161)(138,194,246,160)(139,193,247,159)(140,192,248,158)(141,191,249,157)(142,190,250,156)(143,189,251,155)(144,188,252,154), (1,140,19,122)(2,139,20,121)(3,138,21,120)(4,137,22,119)(5,136,23,118)(6,135,24,117)(7,134,25,116)(8,133,26,115)(9,132,27,114)(10,131,28,113)(11,130,29,112)(12,129,30,111)(13,128,31,110)(14,127,32,109)(15,126,33,144)(16,125,34,143)(17,124,35,142)(18,123,36,141)(37,170,55,152)(38,169,56,151)(39,168,57,150)(40,167,58,149)(41,166,59,148)(42,165,60,147)(43,164,61,146)(44,163,62,145)(45,162,63,180)(46,161,64,179)(47,160,65,178)(48,159,66,177)(49,158,67,176)(50,157,68,175)(51,156,69,174)(52,155,70,173)(53,154,71,172)(54,153,72,171)(73,204,91,186)(74,203,92,185)(75,202,93,184)(76,201,94,183)(77,200,95,182)(78,199,96,181)(79,198,97,216)(80,197,98,215)(81,196,99,214)(82,195,100,213)(83,194,101,212)(84,193,102,211)(85,192,103,210)(86,191,104,209)(87,190,105,208)(88,189,106,207)(89,188,107,206)(90,187,108,205)(217,279,235,261)(218,278,236,260)(219,277,237,259)(220,276,238,258)(221,275,239,257)(222,274,240,256)(223,273,241,255)(224,272,242,254)(225,271,243,253)(226,270,244,288)(227,269,245,287)(228,268,246,286)(229,267,247,285)(230,266,248,284)(231,265,249,283)(232,264,250,282)(233,263,251,281)(234,262,252,280)>;
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), (1,58,266,94)(2,57,267,93)(3,56,268,92)(4,55,269,91)(5,54,270,90)(6,53,271,89)(7,52,272,88)(8,51,273,87)(9,50,274,86)(10,49,275,85)(11,48,276,84)(12,47,277,83)(13,46,278,82)(14,45,279,81)(15,44,280,80)(16,43,281,79)(17,42,282,78)(18,41,283,77)(19,40,284,76)(20,39,285,75)(21,38,286,74)(22,37,287,73)(23,72,288,108)(24,71,253,107)(25,70,254,106)(26,69,255,105)(27,68,256,104)(28,67,257,103)(29,66,258,102)(30,65,259,101)(31,64,260,100)(32,63,261,99)(33,62,262,98)(34,61,263,97)(35,60,264,96)(36,59,265,95)(109,187,217,153)(110,186,218,152)(111,185,219,151)(112,184,220,150)(113,183,221,149)(114,182,222,148)(115,181,223,147)(116,216,224,146)(117,215,225,145)(118,214,226,180)(119,213,227,179)(120,212,228,178)(121,211,229,177)(122,210,230,176)(123,209,231,175)(124,208,232,174)(125,207,233,173)(126,206,234,172)(127,205,235,171)(128,204,236,170)(129,203,237,169)(130,202,238,168)(131,201,239,167)(132,200,240,166)(133,199,241,165)(134,198,242,164)(135,197,243,163)(136,196,244,162)(137,195,245,161)(138,194,246,160)(139,193,247,159)(140,192,248,158)(141,191,249,157)(142,190,250,156)(143,189,251,155)(144,188,252,154), (1,140,19,122)(2,139,20,121)(3,138,21,120)(4,137,22,119)(5,136,23,118)(6,135,24,117)(7,134,25,116)(8,133,26,115)(9,132,27,114)(10,131,28,113)(11,130,29,112)(12,129,30,111)(13,128,31,110)(14,127,32,109)(15,126,33,144)(16,125,34,143)(17,124,35,142)(18,123,36,141)(37,170,55,152)(38,169,56,151)(39,168,57,150)(40,167,58,149)(41,166,59,148)(42,165,60,147)(43,164,61,146)(44,163,62,145)(45,162,63,180)(46,161,64,179)(47,160,65,178)(48,159,66,177)(49,158,67,176)(50,157,68,175)(51,156,69,174)(52,155,70,173)(53,154,71,172)(54,153,72,171)(73,204,91,186)(74,203,92,185)(75,202,93,184)(76,201,94,183)(77,200,95,182)(78,199,96,181)(79,198,97,216)(80,197,98,215)(81,196,99,214)(82,195,100,213)(83,194,101,212)(84,193,102,211)(85,192,103,210)(86,191,104,209)(87,190,105,208)(88,189,106,207)(89,188,107,206)(90,187,108,205)(217,279,235,261)(218,278,236,260)(219,277,237,259)(220,276,238,258)(221,275,239,257)(222,274,240,256)(223,273,241,255)(224,272,242,254)(225,271,243,253)(226,270,244,288)(227,269,245,287)(228,268,246,286)(229,267,247,285)(230,266,248,284)(231,265,249,283)(232,264,250,282)(233,263,251,281)(234,262,252,280) );
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)], [(1,58,266,94),(2,57,267,93),(3,56,268,92),(4,55,269,91),(5,54,270,90),(6,53,271,89),(7,52,272,88),(8,51,273,87),(9,50,274,86),(10,49,275,85),(11,48,276,84),(12,47,277,83),(13,46,278,82),(14,45,279,81),(15,44,280,80),(16,43,281,79),(17,42,282,78),(18,41,283,77),(19,40,284,76),(20,39,285,75),(21,38,286,74),(22,37,287,73),(23,72,288,108),(24,71,253,107),(25,70,254,106),(26,69,255,105),(27,68,256,104),(28,67,257,103),(29,66,258,102),(30,65,259,101),(31,64,260,100),(32,63,261,99),(33,62,262,98),(34,61,263,97),(35,60,264,96),(36,59,265,95),(109,187,217,153),(110,186,218,152),(111,185,219,151),(112,184,220,150),(113,183,221,149),(114,182,222,148),(115,181,223,147),(116,216,224,146),(117,215,225,145),(118,214,226,180),(119,213,227,179),(120,212,228,178),(121,211,229,177),(122,210,230,176),(123,209,231,175),(124,208,232,174),(125,207,233,173),(126,206,234,172),(127,205,235,171),(128,204,236,170),(129,203,237,169),(130,202,238,168),(131,201,239,167),(132,200,240,166),(133,199,241,165),(134,198,242,164),(135,197,243,163),(136,196,244,162),(137,195,245,161),(138,194,246,160),(139,193,247,159),(140,192,248,158),(141,191,249,157),(142,190,250,156),(143,189,251,155),(144,188,252,154)], [(1,140,19,122),(2,139,20,121),(3,138,21,120),(4,137,22,119),(5,136,23,118),(6,135,24,117),(7,134,25,116),(8,133,26,115),(9,132,27,114),(10,131,28,113),(11,130,29,112),(12,129,30,111),(13,128,31,110),(14,127,32,109),(15,126,33,144),(16,125,34,143),(17,124,35,142),(18,123,36,141),(37,170,55,152),(38,169,56,151),(39,168,57,150),(40,167,58,149),(41,166,59,148),(42,165,60,147),(43,164,61,146),(44,163,62,145),(45,162,63,180),(46,161,64,179),(47,160,65,178),(48,159,66,177),(49,158,67,176),(50,157,68,175),(51,156,69,174),(52,155,70,173),(53,154,71,172),(54,153,72,171),(73,204,91,186),(74,203,92,185),(75,202,93,184),(76,201,94,183),(77,200,95,182),(78,199,96,181),(79,198,97,216),(80,197,98,215),(81,196,99,214),(82,195,100,213),(83,194,101,212),(84,193,102,211),(85,192,103,210),(86,191,104,209),(87,190,105,208),(88,189,106,207),(89,188,107,206),(90,187,108,205),(217,279,235,261),(218,278,236,260),(219,277,237,259),(220,276,238,258),(221,275,239,257),(222,274,240,256),(223,273,241,255),(224,272,242,254),(225,271,243,253),(226,270,244,288),(227,269,245,287),(228,268,246,286),(229,267,247,285),(230,266,248,284),(231,265,249,283),(232,264,250,282),(233,263,251,281),(234,262,252,280)]])
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 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | - | + | + | + | - | + | - | ||||||||
| image | C1 | C2 | C2 | C2 | C4 | S3 | D4 | D4 | D6 | SD16 | Q16 | D9 | C4×S3 | C3⋊D4 | D12 | D18 | C24⋊C2 | Dic12 | C4×D9 | C9⋊D4 | D36 | Dic36 | C72⋊C2 |
| kernel | C36.45D4 | C4⋊Dic9 | C2×C72 | C2×Dic18 | Dic18 | C2×C24 | C36 | C2×C18 | C2×C12 | C18 | C18 | C2×C8 | C12 | C12 | C2×C6 | C2×C4 | C6 | C6 | C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 3 | 2 | 2 | 2 | 3 | 4 | 4 | 6 | 6 | 6 | 12 | 12 |
Matrix representation of C36.45D4 ►in GL4(𝔽73) generated by
| 45 | 3 | 0 | 0 |
| 70 | 42 | 0 | 0 |
| 0 | 0 | 29 | 54 |
| 0 | 0 | 19 | 48 |
| 0 | 27 | 0 | 0 |
| 27 | 0 | 0 | 0 |
| 0 | 0 | 20 | 2 |
| 0 | 0 | 55 | 53 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 61 | 10 |
| 0 | 0 | 22 | 12 |
G:=sub<GL(4,GF(73))| [45,70,0,0,3,42,0,0,0,0,29,19,0,0,54,48],[0,27,0,0,27,0,0,0,0,0,20,55,0,0,2,53],[0,1,0,0,1,0,0,0,0,0,61,22,0,0,10,12] >;
C36.45D4 in GAP, Magma, Sage, TeX
C_{36}._{45}D_4 % in TeX
G:=Group("C36.45D4"); // GroupNames label
G:=SmallGroup(288,24);
// by ID
G=gap.SmallGroup(288,24);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,85,92,422,100,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c|a^36=b^4=1,c^2=a^18,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=a^9*b^-1>;
// generators/relations