metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4⋊D27, C27⋊2D8, C4.2D54, C54.8D4, C36.2D6, D108⋊2C2, C12.2D18, C108.2C22, C27⋊C8⋊2C2, C3.(D4⋊D9), C9.(D4⋊S3), (D4×C27)⋊1C2, (C3×D4).2D9, (D4×C9).2S3, C6.17(C9⋊D4), C2.5(C27⋊D4), C18.17(C3⋊D4), SmallGroup(432,16)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4⋊D27
G = < a,b,c,d | a4=b2=c27=d2=1, bab=dad=a-1, ac=ca, bc=cb, dbd=ab, dcd=c-1 >
Smallest permutation representation of D4⋊D27
►On 216 points
Generators in S216
(1 84 51 63)(2 85 52 64)(3 86 53 65)(4 87 54 66)(5 88 28 67)(6 89 29 68)(7 90 30 69)(8 91 31 70)(9 92 32 71)(10 93 33 72)(11 94 34 73)(12 95 35 74)(13 96 36 75)(14 97 37 76)(15 98 38 77)(16 99 39 78)(17 100 40 79)(18 101 41 80)(19 102 42 81)(20 103 43 55)(21 104 44 56)(22 105 45 57)(23 106 46 58)(24 107 47 59)(25 108 48 60)(26 82 49 61)(27 83 50 62)(109 176 143 197)(110 177 144 198)(111 178 145 199)(112 179 146 200)(113 180 147 201)(114 181 148 202)(115 182 149 203)(116 183 150 204)(117 184 151 205)(118 185 152 206)(119 186 153 207)(120 187 154 208)(121 188 155 209)(122 189 156 210)(123 163 157 211)(124 164 158 212)(125 165 159 213)(126 166 160 214)(127 167 161 215)(128 168 162 216)(129 169 136 190)(130 170 137 191)(131 171 138 192)(132 172 139 193)(133 173 140 194)(134 174 141 195)(135 175 142 196)
(1 190)(2 191)(3 192)(4 193)(5 194)(6 195)(7 196)(8 197)(9 198)(10 199)(11 200)(12 201)(13 202)(14 203)(15 204)(16 205)(17 206)(18 207)(19 208)(20 209)(21 210)(22 211)(23 212)(24 213)(25 214)(26 215)(27 216)(28 173)(29 174)(30 175)(31 176)(32 177)(33 178)(34 179)(35 180)(36 181)(37 182)(38 183)(39 184)(40 185)(41 186)(42 187)(43 188)(44 189)(45 163)(46 164)(47 165)(48 166)(49 167)(50 168)(51 169)(52 170)(53 171)(54 172)(55 121)(56 122)(57 123)(58 124)(59 125)(60 126)(61 127)(62 128)(63 129)(64 130)(65 131)(66 132)(67 133)(68 134)(69 135)(70 109)(71 110)(72 111)(73 112)(74 113)(75 114)(76 115)(77 116)(78 117)(79 118)(80 119)(81 120)(82 161)(83 162)(84 136)(85 137)(86 138)(87 139)(88 140)(89 141)(90 142)(91 143)(92 144)(93 145)(94 146)(95 147)(96 148)(97 149)(98 150)(99 151)(100 152)(101 153)(102 154)(103 155)(104 156)(105 157)(106 158)(107 159)(108 160)
(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)
(2 27)(3 26)(4 25)(5 24)(6 23)(7 22)(8 21)(9 20)(10 19)(11 18)(12 17)(13 16)(14 15)(28 47)(29 46)(30 45)(31 44)(32 43)(33 42)(34 41)(35 40)(36 39)(37 38)(48 54)(49 53)(50 52)(55 92)(56 91)(57 90)(58 89)(59 88)(60 87)(61 86)(62 85)(63 84)(64 83)(65 82)(66 108)(67 107)(68 106)(69 105)(70 104)(71 103)(72 102)(73 101)(74 100)(75 99)(76 98)(77 97)(78 96)(79 95)(80 94)(81 93)(109 189)(110 188)(111 187)(112 186)(113 185)(114 184)(115 183)(116 182)(117 181)(118 180)(119 179)(120 178)(121 177)(122 176)(123 175)(124 174)(125 173)(126 172)(127 171)(128 170)(129 169)(130 168)(131 167)(132 166)(133 165)(134 164)(135 163)(136 190)(137 216)(138 215)(139 214)(140 213)(141 212)(142 211)(143 210)(144 209)(145 208)(146 207)(147 206)(148 205)(149 204)(150 203)(151 202)(152 201)(153 200)(154 199)(155 198)(156 197)(157 196)(158 195)(159 194)(160 193)(161 192)(162 191)
G:=sub<Sym(216)| (1,84,51,63)(2,85,52,64)(3,86,53,65)(4,87,54,66)(5,88,28,67)(6,89,29,68)(7,90,30,69)(8,91,31,70)(9,92,32,71)(10,93,33,72)(11,94,34,73)(12,95,35,74)(13,96,36,75)(14,97,37,76)(15,98,38,77)(16,99,39,78)(17,100,40,79)(18,101,41,80)(19,102,42,81)(20,103,43,55)(21,104,44,56)(22,105,45,57)(23,106,46,58)(24,107,47,59)(25,108,48,60)(26,82,49,61)(27,83,50,62)(109,176,143,197)(110,177,144,198)(111,178,145,199)(112,179,146,200)(113,180,147,201)(114,181,148,202)(115,182,149,203)(116,183,150,204)(117,184,151,205)(118,185,152,206)(119,186,153,207)(120,187,154,208)(121,188,155,209)(122,189,156,210)(123,163,157,211)(124,164,158,212)(125,165,159,213)(126,166,160,214)(127,167,161,215)(128,168,162,216)(129,169,136,190)(130,170,137,191)(131,171,138,192)(132,172,139,193)(133,173,140,194)(134,174,141,195)(135,175,142,196), (1,190)(2,191)(3,192)(4,193)(5,194)(6,195)(7,196)(8,197)(9,198)(10,199)(11,200)(12,201)(13,202)(14,203)(15,204)(16,205)(17,206)(18,207)(19,208)(20,209)(21,210)(22,211)(23,212)(24,213)(25,214)(26,215)(27,216)(28,173)(29,174)(30,175)(31,176)(32,177)(33,178)(34,179)(35,180)(36,181)(37,182)(38,183)(39,184)(40,185)(41,186)(42,187)(43,188)(44,189)(45,163)(46,164)(47,165)(48,166)(49,167)(50,168)(51,169)(52,170)(53,171)(54,172)(55,121)(56,122)(57,123)(58,124)(59,125)(60,126)(61,127)(62,128)(63,129)(64,130)(65,131)(66,132)(67,133)(68,134)(69,135)(70,109)(71,110)(72,111)(73,112)(74,113)(75,114)(76,115)(77,116)(78,117)(79,118)(80,119)(81,120)(82,161)(83,162)(84,136)(85,137)(86,138)(87,139)(88,140)(89,141)(90,142)(91,143)(92,144)(93,145)(94,146)(95,147)(96,148)(97,149)(98,150)(99,151)(100,152)(101,153)(102,154)(103,155)(104,156)(105,157)(106,158)(107,159)(108,160), (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), (2,27)(3,26)(4,25)(5,24)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)(28,47)(29,46)(30,45)(31,44)(32,43)(33,42)(34,41)(35,40)(36,39)(37,38)(48,54)(49,53)(50,52)(55,92)(56,91)(57,90)(58,89)(59,88)(60,87)(61,86)(62,85)(63,84)(64,83)(65,82)(66,108)(67,107)(68,106)(69,105)(70,104)(71,103)(72,102)(73,101)(74,100)(75,99)(76,98)(77,97)(78,96)(79,95)(80,94)(81,93)(109,189)(110,188)(111,187)(112,186)(113,185)(114,184)(115,183)(116,182)(117,181)(118,180)(119,179)(120,178)(121,177)(122,176)(123,175)(124,174)(125,173)(126,172)(127,171)(128,170)(129,169)(130,168)(131,167)(132,166)(133,165)(134,164)(135,163)(136,190)(137,216)(138,215)(139,214)(140,213)(141,212)(142,211)(143,210)(144,209)(145,208)(146,207)(147,206)(148,205)(149,204)(150,203)(151,202)(152,201)(153,200)(154,199)(155,198)(156,197)(157,196)(158,195)(159,194)(160,193)(161,192)(162,191)>;
G:=Group( (1,84,51,63)(2,85,52,64)(3,86,53,65)(4,87,54,66)(5,88,28,67)(6,89,29,68)(7,90,30,69)(8,91,31,70)(9,92,32,71)(10,93,33,72)(11,94,34,73)(12,95,35,74)(13,96,36,75)(14,97,37,76)(15,98,38,77)(16,99,39,78)(17,100,40,79)(18,101,41,80)(19,102,42,81)(20,103,43,55)(21,104,44,56)(22,105,45,57)(23,106,46,58)(24,107,47,59)(25,108,48,60)(26,82,49,61)(27,83,50,62)(109,176,143,197)(110,177,144,198)(111,178,145,199)(112,179,146,200)(113,180,147,201)(114,181,148,202)(115,182,149,203)(116,183,150,204)(117,184,151,205)(118,185,152,206)(119,186,153,207)(120,187,154,208)(121,188,155,209)(122,189,156,210)(123,163,157,211)(124,164,158,212)(125,165,159,213)(126,166,160,214)(127,167,161,215)(128,168,162,216)(129,169,136,190)(130,170,137,191)(131,171,138,192)(132,172,139,193)(133,173,140,194)(134,174,141,195)(135,175,142,196), (1,190)(2,191)(3,192)(4,193)(5,194)(6,195)(7,196)(8,197)(9,198)(10,199)(11,200)(12,201)(13,202)(14,203)(15,204)(16,205)(17,206)(18,207)(19,208)(20,209)(21,210)(22,211)(23,212)(24,213)(25,214)(26,215)(27,216)(28,173)(29,174)(30,175)(31,176)(32,177)(33,178)(34,179)(35,180)(36,181)(37,182)(38,183)(39,184)(40,185)(41,186)(42,187)(43,188)(44,189)(45,163)(46,164)(47,165)(48,166)(49,167)(50,168)(51,169)(52,170)(53,171)(54,172)(55,121)(56,122)(57,123)(58,124)(59,125)(60,126)(61,127)(62,128)(63,129)(64,130)(65,131)(66,132)(67,133)(68,134)(69,135)(70,109)(71,110)(72,111)(73,112)(74,113)(75,114)(76,115)(77,116)(78,117)(79,118)(80,119)(81,120)(82,161)(83,162)(84,136)(85,137)(86,138)(87,139)(88,140)(89,141)(90,142)(91,143)(92,144)(93,145)(94,146)(95,147)(96,148)(97,149)(98,150)(99,151)(100,152)(101,153)(102,154)(103,155)(104,156)(105,157)(106,158)(107,159)(108,160), (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), (2,27)(3,26)(4,25)(5,24)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)(28,47)(29,46)(30,45)(31,44)(32,43)(33,42)(34,41)(35,40)(36,39)(37,38)(48,54)(49,53)(50,52)(55,92)(56,91)(57,90)(58,89)(59,88)(60,87)(61,86)(62,85)(63,84)(64,83)(65,82)(66,108)(67,107)(68,106)(69,105)(70,104)(71,103)(72,102)(73,101)(74,100)(75,99)(76,98)(77,97)(78,96)(79,95)(80,94)(81,93)(109,189)(110,188)(111,187)(112,186)(113,185)(114,184)(115,183)(116,182)(117,181)(118,180)(119,179)(120,178)(121,177)(122,176)(123,175)(124,174)(125,173)(126,172)(127,171)(128,170)(129,169)(130,168)(131,167)(132,166)(133,165)(134,164)(135,163)(136,190)(137,216)(138,215)(139,214)(140,213)(141,212)(142,211)(143,210)(144,209)(145,208)(146,207)(147,206)(148,205)(149,204)(150,203)(151,202)(152,201)(153,200)(154,199)(155,198)(156,197)(157,196)(158,195)(159,194)(160,193)(161,192)(162,191) );
G=PermutationGroup([[(1,84,51,63),(2,85,52,64),(3,86,53,65),(4,87,54,66),(5,88,28,67),(6,89,29,68),(7,90,30,69),(8,91,31,70),(9,92,32,71),(10,93,33,72),(11,94,34,73),(12,95,35,74),(13,96,36,75),(14,97,37,76),(15,98,38,77),(16,99,39,78),(17,100,40,79),(18,101,41,80),(19,102,42,81),(20,103,43,55),(21,104,44,56),(22,105,45,57),(23,106,46,58),(24,107,47,59),(25,108,48,60),(26,82,49,61),(27,83,50,62),(109,176,143,197),(110,177,144,198),(111,178,145,199),(112,179,146,200),(113,180,147,201),(114,181,148,202),(115,182,149,203),(116,183,150,204),(117,184,151,205),(118,185,152,206),(119,186,153,207),(120,187,154,208),(121,188,155,209),(122,189,156,210),(123,163,157,211),(124,164,158,212),(125,165,159,213),(126,166,160,214),(127,167,161,215),(128,168,162,216),(129,169,136,190),(130,170,137,191),(131,171,138,192),(132,172,139,193),(133,173,140,194),(134,174,141,195),(135,175,142,196)], [(1,190),(2,191),(3,192),(4,193),(5,194),(6,195),(7,196),(8,197),(9,198),(10,199),(11,200),(12,201),(13,202),(14,203),(15,204),(16,205),(17,206),(18,207),(19,208),(20,209),(21,210),(22,211),(23,212),(24,213),(25,214),(26,215),(27,216),(28,173),(29,174),(30,175),(31,176),(32,177),(33,178),(34,179),(35,180),(36,181),(37,182),(38,183),(39,184),(40,185),(41,186),(42,187),(43,188),(44,189),(45,163),(46,164),(47,165),(48,166),(49,167),(50,168),(51,169),(52,170),(53,171),(54,172),(55,121),(56,122),(57,123),(58,124),(59,125),(60,126),(61,127),(62,128),(63,129),(64,130),(65,131),(66,132),(67,133),(68,134),(69,135),(70,109),(71,110),(72,111),(73,112),(74,113),(75,114),(76,115),(77,116),(78,117),(79,118),(80,119),(81,120),(82,161),(83,162),(84,136),(85,137),(86,138),(87,139),(88,140),(89,141),(90,142),(91,143),(92,144),(93,145),(94,146),(95,147),(96,148),(97,149),(98,150),(99,151),(100,152),(101,153),(102,154),(103,155),(104,156),(105,157),(106,158),(107,159),(108,160)], [(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)], [(2,27),(3,26),(4,25),(5,24),(6,23),(7,22),(8,21),(9,20),(10,19),(11,18),(12,17),(13,16),(14,15),(28,47),(29,46),(30,45),(31,44),(32,43),(33,42),(34,41),(35,40),(36,39),(37,38),(48,54),(49,53),(50,52),(55,92),(56,91),(57,90),(58,89),(59,88),(60,87),(61,86),(62,85),(63,84),(64,83),(65,82),(66,108),(67,107),(68,106),(69,105),(70,104),(71,103),(72,102),(73,101),(74,100),(75,99),(76,98),(77,97),(78,96),(79,95),(80,94),(81,93),(109,189),(110,188),(111,187),(112,186),(113,185),(114,184),(115,183),(116,182),(117,181),(118,180),(119,179),(120,178),(121,177),(122,176),(123,175),(124,174),(125,173),(126,172),(127,171),(128,170),(129,169),(130,168),(131,167),(132,166),(133,165),(134,164),(135,163),(136,190),(137,216),(138,215),(139,214),(140,213),(141,212),(142,211),(143,210),(144,209),(145,208),(146,207),(147,206),(148,205),(149,204),(150,203),(151,202),(152,201),(153,200),(154,199),(155,198),(156,197),(157,196),(158,195),(159,194),(160,193),(161,192),(162,191)]])
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4 | 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 | 2 | 3 | 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 | 108 | 2 | 2 | 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 | D8 | D9 | C3⋊D4 | D18 | D27 | C9⋊D4 | D54 | C27⋊D4 | D4⋊S3 | D4⋊D9 | D4⋊D27 |
| kernel | D4⋊D27 | C27⋊C8 | D108 | 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 | 0 | 1 |
| 0 | 0 | 432 | 0 |
| 432 | 0 | 0 | 0 |
| 0 | 432 | 0 | 0 |
| 0 | 0 | 330 | 103 |
| 0 | 0 | 103 | 103 |
| 401 | 130 | 0 | 0 |
| 303 | 271 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 25 | 195 | 0 | 0 |
| 170 | 408 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 432 |
G:=sub<GL(4,GF(433))| [1,0,0,0,0,1,0,0,0,0,0,432,0,0,1,0],[432,0,0,0,0,432,0,0,0,0,330,103,0,0,103,103],[401,303,0,0,130,271,0,0,0,0,1,0,0,0,0,1],[25,170,0,0,195,408,0,0,0,0,1,0,0,0,0,432] >;
D4⋊D27 in GAP, Magma, Sage, TeX
D_4\rtimes D_{27} % in TeX
G:=Group("D4:D27"); // GroupNames label
G:=SmallGroup(432,16);
// by ID
G=gap.SmallGroup(432,16);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,85,254,135,58,2804,557,10085,292,14118]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^27=d^2=1,b*a*b=d*a*d=a^-1,a*c=c*a,b*c=c*b,d*b*d=a*b,d*c*d=c^-1>;
// generators/relations
Export