metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4.D27, C4.1D54, C36.1D6, C54.7D4, C27⋊2SD16, C12.1D18, Dic54⋊2C2, C108.1C22, C27⋊C8⋊1C2, (D4×C9).1S3, (C3×D4).1D9, C3.(D4.D9), C9.(D4.S3), (D4×C27).1C2, C6.16(C9⋊D4), C2.4(C27⋊D4), C18.16(C3⋊D4), SmallGroup(432,15)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4.D27
G = < a,b,c,d | a4=b2=c27=1, d2=a2, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=c-1 >
Smallest permutation representation of D4.D27
►On 216 points
Generators in S216
(1 107 51 55)(2 108 52 56)(3 82 53 57)(4 83 54 58)(5 84 28 59)(6 85 29 60)(7 86 30 61)(8 87 31 62)(9 88 32 63)(10 89 33 64)(11 90 34 65)(12 91 35 66)(13 92 36 67)(14 93 37 68)(15 94 38 69)(16 95 39 70)(17 96 40 71)(18 97 41 72)(19 98 42 73)(20 99 43 74)(21 100 44 75)(22 101 45 76)(23 102 46 77)(24 103 47 78)(25 104 48 79)(26 105 49 80)(27 106 50 81)(109 171 141 198)(110 172 142 199)(111 173 143 200)(112 174 144 201)(113 175 145 202)(114 176 146 203)(115 177 147 204)(116 178 148 205)(117 179 149 206)(118 180 150 207)(119 181 151 208)(120 182 152 209)(121 183 153 210)(122 184 154 211)(123 185 155 212)(124 186 156 213)(125 187 157 214)(126 188 158 215)(127 189 159 216)(128 163 160 190)(129 164 161 191)(130 165 162 192)(131 166 136 193)(132 167 137 194)(133 168 138 195)(134 169 139 196)(135 170 140 197)
(1 55)(2 56)(3 57)(4 58)(5 59)(6 60)(7 61)(8 62)(9 63)(10 64)(11 65)(12 66)(13 67)(14 68)(15 69)(16 70)(17 71)(18 72)(19 73)(20 74)(21 75)(22 76)(23 77)(24 78)(25 79)(26 80)(27 81)(28 84)(29 85)(30 86)(31 87)(32 88)(33 89)(34 90)(35 91)(36 92)(37 93)(38 94)(39 95)(40 96)(41 97)(42 98)(43 99)(44 100)(45 101)(46 102)(47 103)(48 104)(49 105)(50 106)(51 107)(52 108)(53 82)(54 83)(109 141)(110 142)(111 143)(112 144)(113 145)(114 146)(115 147)(116 148)(117 149)(118 150)(119 151)(120 152)(121 153)(122 154)(123 155)(124 156)(125 157)(126 158)(127 159)(128 160)(129 161)(130 162)(131 136)(132 137)(133 138)(134 139)(135 140)
(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)
(1 149 51 117)(2 148 52 116)(3 147 53 115)(4 146 54 114)(5 145 28 113)(6 144 29 112)(7 143 30 111)(8 142 31 110)(9 141 32 109)(10 140 33 135)(11 139 34 134)(12 138 35 133)(13 137 36 132)(14 136 37 131)(15 162 38 130)(16 161 39 129)(17 160 40 128)(18 159 41 127)(19 158 42 126)(20 157 43 125)(21 156 44 124)(22 155 45 123)(23 154 46 122)(24 153 47 121)(25 152 48 120)(26 151 49 119)(27 150 50 118)(55 206 107 179)(56 205 108 178)(57 204 82 177)(58 203 83 176)(59 202 84 175)(60 201 85 174)(61 200 86 173)(62 199 87 172)(63 198 88 171)(64 197 89 170)(65 196 90 169)(66 195 91 168)(67 194 92 167)(68 193 93 166)(69 192 94 165)(70 191 95 164)(71 190 96 163)(72 216 97 189)(73 215 98 188)(74 214 99 187)(75 213 100 186)(76 212 101 185)(77 211 102 184)(78 210 103 183)(79 209 104 182)(80 208 105 181)(81 207 106 180)
G:=sub<Sym(216)| (1,107,51,55)(2,108,52,56)(3,82,53,57)(4,83,54,58)(5,84,28,59)(6,85,29,60)(7,86,30,61)(8,87,31,62)(9,88,32,63)(10,89,33,64)(11,90,34,65)(12,91,35,66)(13,92,36,67)(14,93,37,68)(15,94,38,69)(16,95,39,70)(17,96,40,71)(18,97,41,72)(19,98,42,73)(20,99,43,74)(21,100,44,75)(22,101,45,76)(23,102,46,77)(24,103,47,78)(25,104,48,79)(26,105,49,80)(27,106,50,81)(109,171,141,198)(110,172,142,199)(111,173,143,200)(112,174,144,201)(113,175,145,202)(114,176,146,203)(115,177,147,204)(116,178,148,205)(117,179,149,206)(118,180,150,207)(119,181,151,208)(120,182,152,209)(121,183,153,210)(122,184,154,211)(123,185,155,212)(124,186,156,213)(125,187,157,214)(126,188,158,215)(127,189,159,216)(128,163,160,190)(129,164,161,191)(130,165,162,192)(131,166,136,193)(132,167,137,194)(133,168,138,195)(134,169,139,196)(135,170,140,197), (1,55)(2,56)(3,57)(4,58)(5,59)(6,60)(7,61)(8,62)(9,63)(10,64)(11,65)(12,66)(13,67)(14,68)(15,69)(16,70)(17,71)(18,72)(19,73)(20,74)(21,75)(22,76)(23,77)(24,78)(25,79)(26,80)(27,81)(28,84)(29,85)(30,86)(31,87)(32,88)(33,89)(34,90)(35,91)(36,92)(37,93)(38,94)(39,95)(40,96)(41,97)(42,98)(43,99)(44,100)(45,101)(46,102)(47,103)(48,104)(49,105)(50,106)(51,107)(52,108)(53,82)(54,83)(109,141)(110,142)(111,143)(112,144)(113,145)(114,146)(115,147)(116,148)(117,149)(118,150)(119,151)(120,152)(121,153)(122,154)(123,155)(124,156)(125,157)(126,158)(127,159)(128,160)(129,161)(130,162)(131,136)(132,137)(133,138)(134,139)(135,140), (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), (1,149,51,117)(2,148,52,116)(3,147,53,115)(4,146,54,114)(5,145,28,113)(6,144,29,112)(7,143,30,111)(8,142,31,110)(9,141,32,109)(10,140,33,135)(11,139,34,134)(12,138,35,133)(13,137,36,132)(14,136,37,131)(15,162,38,130)(16,161,39,129)(17,160,40,128)(18,159,41,127)(19,158,42,126)(20,157,43,125)(21,156,44,124)(22,155,45,123)(23,154,46,122)(24,153,47,121)(25,152,48,120)(26,151,49,119)(27,150,50,118)(55,206,107,179)(56,205,108,178)(57,204,82,177)(58,203,83,176)(59,202,84,175)(60,201,85,174)(61,200,86,173)(62,199,87,172)(63,198,88,171)(64,197,89,170)(65,196,90,169)(66,195,91,168)(67,194,92,167)(68,193,93,166)(69,192,94,165)(70,191,95,164)(71,190,96,163)(72,216,97,189)(73,215,98,188)(74,214,99,187)(75,213,100,186)(76,212,101,185)(77,211,102,184)(78,210,103,183)(79,209,104,182)(80,208,105,181)(81,207,106,180)>;
G:=Group( (1,107,51,55)(2,108,52,56)(3,82,53,57)(4,83,54,58)(5,84,28,59)(6,85,29,60)(7,86,30,61)(8,87,31,62)(9,88,32,63)(10,89,33,64)(11,90,34,65)(12,91,35,66)(13,92,36,67)(14,93,37,68)(15,94,38,69)(16,95,39,70)(17,96,40,71)(18,97,41,72)(19,98,42,73)(20,99,43,74)(21,100,44,75)(22,101,45,76)(23,102,46,77)(24,103,47,78)(25,104,48,79)(26,105,49,80)(27,106,50,81)(109,171,141,198)(110,172,142,199)(111,173,143,200)(112,174,144,201)(113,175,145,202)(114,176,146,203)(115,177,147,204)(116,178,148,205)(117,179,149,206)(118,180,150,207)(119,181,151,208)(120,182,152,209)(121,183,153,210)(122,184,154,211)(123,185,155,212)(124,186,156,213)(125,187,157,214)(126,188,158,215)(127,189,159,216)(128,163,160,190)(129,164,161,191)(130,165,162,192)(131,166,136,193)(132,167,137,194)(133,168,138,195)(134,169,139,196)(135,170,140,197), (1,55)(2,56)(3,57)(4,58)(5,59)(6,60)(7,61)(8,62)(9,63)(10,64)(11,65)(12,66)(13,67)(14,68)(15,69)(16,70)(17,71)(18,72)(19,73)(20,74)(21,75)(22,76)(23,77)(24,78)(25,79)(26,80)(27,81)(28,84)(29,85)(30,86)(31,87)(32,88)(33,89)(34,90)(35,91)(36,92)(37,93)(38,94)(39,95)(40,96)(41,97)(42,98)(43,99)(44,100)(45,101)(46,102)(47,103)(48,104)(49,105)(50,106)(51,107)(52,108)(53,82)(54,83)(109,141)(110,142)(111,143)(112,144)(113,145)(114,146)(115,147)(116,148)(117,149)(118,150)(119,151)(120,152)(121,153)(122,154)(123,155)(124,156)(125,157)(126,158)(127,159)(128,160)(129,161)(130,162)(131,136)(132,137)(133,138)(134,139)(135,140), (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), (1,149,51,117)(2,148,52,116)(3,147,53,115)(4,146,54,114)(5,145,28,113)(6,144,29,112)(7,143,30,111)(8,142,31,110)(9,141,32,109)(10,140,33,135)(11,139,34,134)(12,138,35,133)(13,137,36,132)(14,136,37,131)(15,162,38,130)(16,161,39,129)(17,160,40,128)(18,159,41,127)(19,158,42,126)(20,157,43,125)(21,156,44,124)(22,155,45,123)(23,154,46,122)(24,153,47,121)(25,152,48,120)(26,151,49,119)(27,150,50,118)(55,206,107,179)(56,205,108,178)(57,204,82,177)(58,203,83,176)(59,202,84,175)(60,201,85,174)(61,200,86,173)(62,199,87,172)(63,198,88,171)(64,197,89,170)(65,196,90,169)(66,195,91,168)(67,194,92,167)(68,193,93,166)(69,192,94,165)(70,191,95,164)(71,190,96,163)(72,216,97,189)(73,215,98,188)(74,214,99,187)(75,213,100,186)(76,212,101,185)(77,211,102,184)(78,210,103,183)(79,209,104,182)(80,208,105,181)(81,207,106,180) );
G=PermutationGroup([[(1,107,51,55),(2,108,52,56),(3,82,53,57),(4,83,54,58),(5,84,28,59),(6,85,29,60),(7,86,30,61),(8,87,31,62),(9,88,32,63),(10,89,33,64),(11,90,34,65),(12,91,35,66),(13,92,36,67),(14,93,37,68),(15,94,38,69),(16,95,39,70),(17,96,40,71),(18,97,41,72),(19,98,42,73),(20,99,43,74),(21,100,44,75),(22,101,45,76),(23,102,46,77),(24,103,47,78),(25,104,48,79),(26,105,49,80),(27,106,50,81),(109,171,141,198),(110,172,142,199),(111,173,143,200),(112,174,144,201),(113,175,145,202),(114,176,146,203),(115,177,147,204),(116,178,148,205),(117,179,149,206),(118,180,150,207),(119,181,151,208),(120,182,152,209),(121,183,153,210),(122,184,154,211),(123,185,155,212),(124,186,156,213),(125,187,157,214),(126,188,158,215),(127,189,159,216),(128,163,160,190),(129,164,161,191),(130,165,162,192),(131,166,136,193),(132,167,137,194),(133,168,138,195),(134,169,139,196),(135,170,140,197)], [(1,55),(2,56),(3,57),(4,58),(5,59),(6,60),(7,61),(8,62),(9,63),(10,64),(11,65),(12,66),(13,67),(14,68),(15,69),(16,70),(17,71),(18,72),(19,73),(20,74),(21,75),(22,76),(23,77),(24,78),(25,79),(26,80),(27,81),(28,84),(29,85),(30,86),(31,87),(32,88),(33,89),(34,90),(35,91),(36,92),(37,93),(38,94),(39,95),(40,96),(41,97),(42,98),(43,99),(44,100),(45,101),(46,102),(47,103),(48,104),(49,105),(50,106),(51,107),(52,108),(53,82),(54,83),(109,141),(110,142),(111,143),(112,144),(113,145),(114,146),(115,147),(116,148),(117,149),(118,150),(119,151),(120,152),(121,153),(122,154),(123,155),(124,156),(125,157),(126,158),(127,159),(128,160),(129,161),(130,162),(131,136),(132,137),(133,138),(134,139),(135,140)], [(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)], [(1,149,51,117),(2,148,52,116),(3,147,53,115),(4,146,54,114),(5,145,28,113),(6,144,29,112),(7,143,30,111),(8,142,31,110),(9,141,32,109),(10,140,33,135),(11,139,34,134),(12,138,35,133),(13,137,36,132),(14,136,37,131),(15,162,38,130),(16,161,39,129),(17,160,40,128),(18,159,41,127),(19,158,42,126),(20,157,43,125),(21,156,44,124),(22,155,45,123),(23,154,46,122),(24,153,47,121),(25,152,48,120),(26,151,49,119),(27,150,50,118),(55,206,107,179),(56,205,108,178),(57,204,82,177),(58,203,83,176),(59,202,84,175),(60,201,85,174),(61,200,86,173),(62,199,87,172),(63,198,88,171),(64,197,89,170),(65,196,90,169),(66,195,91,168),(67,194,92,167),(68,193,93,166),(69,192,94,165),(70,191,95,164),(71,190,96,163),(72,216,97,189),(73,215,98,188),(74,214,99,187),(75,213,100,186),(76,212,101,185),(77,211,102,184),(78,210,103,183),(79,209,104,182),(80,208,105,181),(81,207,106,180)]])
72 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4A | 4B | 6A | 6B | 6C | 8A | 8B | 9A | 9B | 9C | 12 | 18A | 18B | 18C | 18D | ··· | 18I | 27A | ··· | 27I | 36A | 36B | 36C | 54A | ··· | 54I | 54J | ··· | 54AA | 108A | ··· | 108I |
| order | 1 | 2 | 2 | 3 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 9 | 9 | 9 | 12 | 18 | 18 | 18 | 18 | ··· | 18 | 27 | ··· | 27 | 36 | 36 | 36 | 54 | ··· | 54 | 54 | ··· | 54 | 108 | ··· | 108 |
| size | 1 | 1 | 4 | 2 | 2 | 108 | 2 | 4 | 4 | 54 | 54 | 2 | 2 | 2 | 4 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | - | - | ||||
| image | C1 | C2 | C2 | C2 | S3 | D4 | D6 | SD16 | D9 | C3⋊D4 | D18 | D27 | C9⋊D4 | D54 | C27⋊D4 | D4.S3 | D4.D9 | D4.D27 |
| kernel | D4.D27 | C27⋊C8 | Dic54 | D4×C27 | D4×C9 | C54 | C36 | C27 | C3×D4 | C18 | C12 | D4 | C6 | C4 | C2 | C9 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 3 | 2 | 3 | 9 | 6 | 9 | 18 | 1 | 3 | 9 |
Matrix representation of D4.D27 ►in GL4(𝔽433) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 273 | 164 |
| 0 | 0 | 100 | 160 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 273 | 164 |
| 0 | 0 | 401 | 160 |
| 25 | 170 | 0 | 0 |
| 263 | 195 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 84 | 311 | 0 | 0 |
| 395 | 349 | 0 | 0 |
| 0 | 0 | 291 | 29 |
| 0 | 0 | 96 | 142 |
G:=sub<GL(4,GF(433))| [1,0,0,0,0,1,0,0,0,0,273,100,0,0,164,160],[1,0,0,0,0,1,0,0,0,0,273,401,0,0,164,160],[25,263,0,0,170,195,0,0,0,0,1,0,0,0,0,1],[84,395,0,0,311,349,0,0,0,0,291,96,0,0,29,142] >;
D4.D27 in GAP, Magma, Sage, TeX
D_4.D_{27} % in TeX
G:=Group("D4.D27"); // GroupNames label
G:=SmallGroup(432,15);
// by ID
G=gap.SmallGroup(432,15);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,85,254,135,58,2804,557,10085,292,14118]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^27=1,d^2=a^2,b*a*b=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a*b,d*c*d^-1=c^-1>;
// generators/relations
Export