metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4⋊2D27, C36.6D6, C4.5D54, C12.6D18, Dic54⋊3C2, C54.6C23, C22.1D54, C108.5C22, D54.2C22, Dic27.3C22, (D4×C27)⋊3C2, (C4×D27)⋊2C2, C27⋊2(C4○D4), C27⋊D4⋊2C2, (C3×D4).4D9, (D4×C9).4S3, (C2×C18).3D6, (C2×C6).3D18, (C2×C54).C22, C9.(D4⋊2S3), C3.(D4⋊2D9), (C2×Dic27)⋊3C2, C6.33(C22×D9), C2.7(C22×D27), C18.33(C22×S3), SmallGroup(432,48)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4⋊2D27
G = < a,b,c,d | a4=b2=c27=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a2b, dcd=c-1 >
Subgroups: 560 in 80 conjugacy classes, 35 normal (21 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, D4, D4, Q8, C9, Dic3, C12, D6, C2×C6, C4○D4, D9, C18, C18, Dic6, C4×S3, C2×Dic3, C3⋊D4, C3×D4, C27, Dic9, C36, D18, C2×C18, D4⋊2S3, D27, C54, C54, Dic18, C4×D9, C2×Dic9, C9⋊D4, D4×C9, Dic27, Dic27, C108, D54, C2×C54, D4⋊2D9, Dic54, C4×D27, C2×Dic27, C27⋊D4, D4×C27, D4⋊2D27
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, D9, C22×S3, D18, D4⋊2S3, D27, C22×D9, D54, D4⋊2D9, C22×D27, D4⋊2D27
►On 216 points
Generators in S216
(1 55 54 96)(2 56 28 97)(3 57 29 98)(4 58 30 99)(5 59 31 100)(6 60 32 101)(7 61 33 102)(8 62 34 103)(9 63 35 104)(10 64 36 105)(11 65 37 106)(12 66 38 107)(13 67 39 108)(14 68 40 82)(15 69 41 83)(16 70 42 84)(17 71 43 85)(18 72 44 86)(19 73 45 87)(20 74 46 88)(21 75 47 89)(22 76 48 90)(23 77 49 91)(24 78 50 92)(25 79 51 93)(26 80 52 94)(27 81 53 95)(109 163 144 206)(110 164 145 207)(111 165 146 208)(112 166 147 209)(113 167 148 210)(114 168 149 211)(115 169 150 212)(116 170 151 213)(117 171 152 214)(118 172 153 215)(119 173 154 216)(120 174 155 190)(121 175 156 191)(122 176 157 192)(123 177 158 193)(124 178 159 194)(125 179 160 195)(126 180 161 196)(127 181 162 197)(128 182 136 198)(129 183 137 199)(130 184 138 200)(131 185 139 201)(132 186 140 202)(133 187 141 203)(134 188 142 204)(135 189 143 205)
(55 96)(56 97)(57 98)(58 99)(59 100)(60 101)(61 102)(62 103)(63 104)(64 105)(65 106)(66 107)(67 108)(68 82)(69 83)(70 84)(71 85)(72 86)(73 87)(74 88)(75 89)(76 90)(77 91)(78 92)(79 93)(80 94)(81 95)(109 144)(110 145)(111 146)(112 147)(113 148)(114 149)(115 150)(116 151)(117 152)(118 153)(119 154)(120 155)(121 156)(122 157)(123 158)(124 159)(125 160)(126 161)(127 162)(128 136)(129 137)(130 138)(131 139)(132 140)(133 141)(134 142)(135 143)
(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)
(1 112)(2 111)(3 110)(4 109)(5 135)(6 134)(7 133)(8 132)(9 131)(10 130)(11 129)(12 128)(13 127)(14 126)(15 125)(16 124)(17 123)(18 122)(19 121)(20 120)(21 119)(22 118)(23 117)(24 116)(25 115)(26 114)(27 113)(28 146)(29 145)(30 144)(31 143)(32 142)(33 141)(34 140)(35 139)(36 138)(37 137)(38 136)(39 162)(40 161)(41 160)(42 159)(43 158)(44 157)(45 156)(46 155)(47 154)(48 153)(49 152)(50 151)(51 150)(52 149)(53 148)(54 147)(55 166)(56 165)(57 164)(58 163)(59 189)(60 188)(61 187)(62 186)(63 185)(64 184)(65 183)(66 182)(67 181)(68 180)(69 179)(70 178)(71 177)(72 176)(73 175)(74 174)(75 173)(76 172)(77 171)(78 170)(79 169)(80 168)(81 167)(82 196)(83 195)(84 194)(85 193)(86 192)(87 191)(88 190)(89 216)(90 215)(91 214)(92 213)(93 212)(94 211)(95 210)(96 209)(97 208)(98 207)(99 206)(100 205)(101 204)(102 203)(103 202)(104 201)(105 200)(106 199)(107 198)(108 197)
G:=sub<Sym(216)| (1,55,54,96)(2,56,28,97)(3,57,29,98)(4,58,30,99)(5,59,31,100)(6,60,32,101)(7,61,33,102)(8,62,34,103)(9,63,35,104)(10,64,36,105)(11,65,37,106)(12,66,38,107)(13,67,39,108)(14,68,40,82)(15,69,41,83)(16,70,42,84)(17,71,43,85)(18,72,44,86)(19,73,45,87)(20,74,46,88)(21,75,47,89)(22,76,48,90)(23,77,49,91)(24,78,50,92)(25,79,51,93)(26,80,52,94)(27,81,53,95)(109,163,144,206)(110,164,145,207)(111,165,146,208)(112,166,147,209)(113,167,148,210)(114,168,149,211)(115,169,150,212)(116,170,151,213)(117,171,152,214)(118,172,153,215)(119,173,154,216)(120,174,155,190)(121,175,156,191)(122,176,157,192)(123,177,158,193)(124,178,159,194)(125,179,160,195)(126,180,161,196)(127,181,162,197)(128,182,136,198)(129,183,137,199)(130,184,138,200)(131,185,139,201)(132,186,140,202)(133,187,141,203)(134,188,142,204)(135,189,143,205), (55,96)(56,97)(57,98)(58,99)(59,100)(60,101)(61,102)(62,103)(63,104)(64,105)(65,106)(66,107)(67,108)(68,82)(69,83)(70,84)(71,85)(72,86)(73,87)(74,88)(75,89)(76,90)(77,91)(78,92)(79,93)(80,94)(81,95)(109,144)(110,145)(111,146)(112,147)(113,148)(114,149)(115,150)(116,151)(117,152)(118,153)(119,154)(120,155)(121,156)(122,157)(123,158)(124,159)(125,160)(126,161)(127,162)(128,136)(129,137)(130,138)(131,139)(132,140)(133,141)(134,142)(135,143), (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), (1,112)(2,111)(3,110)(4,109)(5,135)(6,134)(7,133)(8,132)(9,131)(10,130)(11,129)(12,128)(13,127)(14,126)(15,125)(16,124)(17,123)(18,122)(19,121)(20,120)(21,119)(22,118)(23,117)(24,116)(25,115)(26,114)(27,113)(28,146)(29,145)(30,144)(31,143)(32,142)(33,141)(34,140)(35,139)(36,138)(37,137)(38,136)(39,162)(40,161)(41,160)(42,159)(43,158)(44,157)(45,156)(46,155)(47,154)(48,153)(49,152)(50,151)(51,150)(52,149)(53,148)(54,147)(55,166)(56,165)(57,164)(58,163)(59,189)(60,188)(61,187)(62,186)(63,185)(64,184)(65,183)(66,182)(67,181)(68,180)(69,179)(70,178)(71,177)(72,176)(73,175)(74,174)(75,173)(76,172)(77,171)(78,170)(79,169)(80,168)(81,167)(82,196)(83,195)(84,194)(85,193)(86,192)(87,191)(88,190)(89,216)(90,215)(91,214)(92,213)(93,212)(94,211)(95,210)(96,209)(97,208)(98,207)(99,206)(100,205)(101,204)(102,203)(103,202)(104,201)(105,200)(106,199)(107,198)(108,197)>;
G:=Group( (1,55,54,96)(2,56,28,97)(3,57,29,98)(4,58,30,99)(5,59,31,100)(6,60,32,101)(7,61,33,102)(8,62,34,103)(9,63,35,104)(10,64,36,105)(11,65,37,106)(12,66,38,107)(13,67,39,108)(14,68,40,82)(15,69,41,83)(16,70,42,84)(17,71,43,85)(18,72,44,86)(19,73,45,87)(20,74,46,88)(21,75,47,89)(22,76,48,90)(23,77,49,91)(24,78,50,92)(25,79,51,93)(26,80,52,94)(27,81,53,95)(109,163,144,206)(110,164,145,207)(111,165,146,208)(112,166,147,209)(113,167,148,210)(114,168,149,211)(115,169,150,212)(116,170,151,213)(117,171,152,214)(118,172,153,215)(119,173,154,216)(120,174,155,190)(121,175,156,191)(122,176,157,192)(123,177,158,193)(124,178,159,194)(125,179,160,195)(126,180,161,196)(127,181,162,197)(128,182,136,198)(129,183,137,199)(130,184,138,200)(131,185,139,201)(132,186,140,202)(133,187,141,203)(134,188,142,204)(135,189,143,205), (55,96)(56,97)(57,98)(58,99)(59,100)(60,101)(61,102)(62,103)(63,104)(64,105)(65,106)(66,107)(67,108)(68,82)(69,83)(70,84)(71,85)(72,86)(73,87)(74,88)(75,89)(76,90)(77,91)(78,92)(79,93)(80,94)(81,95)(109,144)(110,145)(111,146)(112,147)(113,148)(114,149)(115,150)(116,151)(117,152)(118,153)(119,154)(120,155)(121,156)(122,157)(123,158)(124,159)(125,160)(126,161)(127,162)(128,136)(129,137)(130,138)(131,139)(132,140)(133,141)(134,142)(135,143), (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), (1,112)(2,111)(3,110)(4,109)(5,135)(6,134)(7,133)(8,132)(9,131)(10,130)(11,129)(12,128)(13,127)(14,126)(15,125)(16,124)(17,123)(18,122)(19,121)(20,120)(21,119)(22,118)(23,117)(24,116)(25,115)(26,114)(27,113)(28,146)(29,145)(30,144)(31,143)(32,142)(33,141)(34,140)(35,139)(36,138)(37,137)(38,136)(39,162)(40,161)(41,160)(42,159)(43,158)(44,157)(45,156)(46,155)(47,154)(48,153)(49,152)(50,151)(51,150)(52,149)(53,148)(54,147)(55,166)(56,165)(57,164)(58,163)(59,189)(60,188)(61,187)(62,186)(63,185)(64,184)(65,183)(66,182)(67,181)(68,180)(69,179)(70,178)(71,177)(72,176)(73,175)(74,174)(75,173)(76,172)(77,171)(78,170)(79,169)(80,168)(81,167)(82,196)(83,195)(84,194)(85,193)(86,192)(87,191)(88,190)(89,216)(90,215)(91,214)(92,213)(93,212)(94,211)(95,210)(96,209)(97,208)(98,207)(99,206)(100,205)(101,204)(102,203)(103,202)(104,201)(105,200)(106,199)(107,198)(108,197) );
G=PermutationGroup([[(1,55,54,96),(2,56,28,97),(3,57,29,98),(4,58,30,99),(5,59,31,100),(6,60,32,101),(7,61,33,102),(8,62,34,103),(9,63,35,104),(10,64,36,105),(11,65,37,106),(12,66,38,107),(13,67,39,108),(14,68,40,82),(15,69,41,83),(16,70,42,84),(17,71,43,85),(18,72,44,86),(19,73,45,87),(20,74,46,88),(21,75,47,89),(22,76,48,90),(23,77,49,91),(24,78,50,92),(25,79,51,93),(26,80,52,94),(27,81,53,95),(109,163,144,206),(110,164,145,207),(111,165,146,208),(112,166,147,209),(113,167,148,210),(114,168,149,211),(115,169,150,212),(116,170,151,213),(117,171,152,214),(118,172,153,215),(119,173,154,216),(120,174,155,190),(121,175,156,191),(122,176,157,192),(123,177,158,193),(124,178,159,194),(125,179,160,195),(126,180,161,196),(127,181,162,197),(128,182,136,198),(129,183,137,199),(130,184,138,200),(131,185,139,201),(132,186,140,202),(133,187,141,203),(134,188,142,204),(135,189,143,205)], [(55,96),(56,97),(57,98),(58,99),(59,100),(60,101),(61,102),(62,103),(63,104),(64,105),(65,106),(66,107),(67,108),(68,82),(69,83),(70,84),(71,85),(72,86),(73,87),(74,88),(75,89),(76,90),(77,91),(78,92),(79,93),(80,94),(81,95),(109,144),(110,145),(111,146),(112,147),(113,148),(114,149),(115,150),(116,151),(117,152),(118,153),(119,154),(120,155),(121,156),(122,157),(123,158),(124,159),(125,160),(126,161),(127,162),(128,136),(129,137),(130,138),(131,139),(132,140),(133,141),(134,142),(135,143)], [(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)], [(1,112),(2,111),(3,110),(4,109),(5,135),(6,134),(7,133),(8,132),(9,131),(10,130),(11,129),(12,128),(13,127),(14,126),(15,125),(16,124),(17,123),(18,122),(19,121),(20,120),(21,119),(22,118),(23,117),(24,116),(25,115),(26,114),(27,113),(28,146),(29,145),(30,144),(31,143),(32,142),(33,141),(34,140),(35,139),(36,138),(37,137),(38,136),(39,162),(40,161),(41,160),(42,159),(43,158),(44,157),(45,156),(46,155),(47,154),(48,153),(49,152),(50,151),(51,150),(52,149),(53,148),(54,147),(55,166),(56,165),(57,164),(58,163),(59,189),(60,188),(61,187),(62,186),(63,185),(64,184),(65,183),(66,182),(67,181),(68,180),(69,179),(70,178),(71,177),(72,176),(73,175),(74,174),(75,173),(76,172),(77,171),(78,170),(79,169),(80,168),(81,167),(82,196),(83,195),(84,194),(85,193),(86,192),(87,191),(88,190),(89,216),(90,215),(91,214),(92,213),(93,212),(94,211),(95,210),(96,209),(97,208),(98,207),(99,206),(100,205),(101,204),(102,203),(103,202),(104,201),(105,200),(106,199),(107,198),(108,197)]])
75 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 9A | 9B | 9C | 12 | 18A | 18B | 18C | 18D | ··· | 18I | 27A | ··· | 27I | 36A | 36B | 36C | 54A | ··· | 54I | 54J | ··· | 54AA | 108A | ··· | 108I |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 9 | 9 | 9 | 12 | 18 | 18 | 18 | 18 | ··· | 18 | 27 | ··· | 27 | 36 | 36 | 36 | 54 | ··· | 54 | 54 | ··· | 54 | 108 | ··· | 108 |
| size | 1 | 1 | 2 | 2 | 54 | 2 | 2 | 27 | 27 | 54 | 54 | 2 | 4 | 4 | 2 | 2 | 2 | 4 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
75 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | C4○D4 | D9 | D18 | D18 | D27 | D54 | D54 | D4⋊2S3 | D4⋊2D9 | D4⋊2D27 |
| kernel | D4⋊2D27 | Dic54 | C4×D27 | C2×Dic27 | C27⋊D4 | D4×C27 | D4×C9 | C36 | C2×C18 | C27 | C3×D4 | C12 | C2×C6 | D4 | C4 | C22 | C9 | C3 | C1 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 3 | 3 | 6 | 9 | 9 | 18 | 1 | 3 | 9 |
Matrix representation of D4⋊2D27 ►in GL4(𝔽109) generated by
| 108 | 0 | 0 | 0 |
| 0 | 108 | 0 | 0 |
| 0 | 0 | 108 | 108 |
| 0 | 0 | 2 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 107 | 108 |
| 10 | 102 | 0 | 0 |
| 7 | 17 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 80 | 58 | 0 | 0 |
| 87 | 29 | 0 | 0 |
| 0 | 0 | 33 | 33 |
| 0 | 0 | 43 | 76 |
G:=sub<GL(4,GF(109))| [108,0,0,0,0,108,0,0,0,0,108,2,0,0,108,1],[1,0,0,0,0,1,0,0,0,0,1,107,0,0,0,108],[10,7,0,0,102,17,0,0,0,0,1,0,0,0,0,1],[80,87,0,0,58,29,0,0,0,0,33,43,0,0,33,76] >;
D4⋊2D27 in GAP, Magma, Sage, TeX
D_4\rtimes_2D_{27} % in TeX
G:=Group("D4:2D27"); // GroupNames label
G:=SmallGroup(432,48);
// by ID
G=gap.SmallGroup(432,48);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,64,254,135,2804,557,10085,292,14118]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^27=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations