direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4×D27, C4⋊1D54, C108⋊C22, C36.5D6, D108⋊3C2, C12.5D18, C22⋊2D54, D54⋊2C22, C54.5C23, Dic27⋊1C22, C3.(D4×D9), C9.(S3×D4), C27⋊2(C2×D4), (C2×C54)⋊C22, (D4×C27)⋊2C2, (C4×D27)⋊1C2, C27⋊D4⋊1C2, (D4×C9).3S3, (C3×D4).3D9, (C2×C18).2D6, (C2×C6).2D18, (C22×D27)⋊2C2, C6.32(C22×D9), C2.6(C22×D27), C18.32(C22×S3), SmallGroup(432,47)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4×D27
G = < a,b,c,d | a4=b2=c27=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 1040 in 108 conjugacy classes, 37 normal (21 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, D4, D4, C23, C9, Dic3, C12, D6, C2×C6, C2×D4, D9, C18, C18, C4×S3, D12, C3⋊D4, C3×D4, C22×S3, C27, Dic9, C36, D18, C2×C18, S3×D4, D27, D27, C54, C54, C4×D9, D36, C9⋊D4, D4×C9, C22×D9, Dic27, C108, D54, D54, D54, C2×C54, D4×D9, C4×D27, D108, C27⋊D4, D4×C27, C22×D27, D4×D27
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D9, C22×S3, D18, S3×D4, D27, C22×D9, D54, D4×D9, C22×D27, D4×D27
►On 108 points
Generators in S108
(1 94 29 65)(2 95 30 66)(3 96 31 67)(4 97 32 68)(5 98 33 69)(6 99 34 70)(7 100 35 71)(8 101 36 72)(9 102 37 73)(10 103 38 74)(11 104 39 75)(12 105 40 76)(13 106 41 77)(14 107 42 78)(15 108 43 79)(16 82 44 80)(17 83 45 81)(18 84 46 55)(19 85 47 56)(20 86 48 57)(21 87 49 58)(22 88 50 59)(23 89 51 60)(24 90 52 61)(25 91 53 62)(26 92 54 63)(27 93 28 64)
(1 65)(2 66)(3 67)(4 68)(5 69)(6 70)(7 71)(8 72)(9 73)(10 74)(11 75)(12 76)(13 77)(14 78)(15 79)(16 80)(17 81)(18 55)(19 56)(20 57)(21 58)(22 59)(23 60)(24 61)(25 62)(26 63)(27 64)(28 93)(29 94)(30 95)(31 96)(32 97)(33 98)(34 99)(35 100)(36 101)(37 102)(38 103)(39 104)(40 105)(41 106)(42 107)(43 108)(44 82)(45 83)(46 84)(47 85)(48 86)(49 87)(50 88)(51 89)(52 90)(53 91)(54 92)
(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)
(1 29)(2 28)(3 54)(4 53)(5 52)(6 51)(7 50)(8 49)(9 48)(10 47)(11 46)(12 45)(13 44)(14 43)(15 42)(16 41)(17 40)(18 39)(19 38)(20 37)(21 36)(22 35)(23 34)(24 33)(25 32)(26 31)(27 30)(55 104)(56 103)(57 102)(58 101)(59 100)(60 99)(61 98)(62 97)(63 96)(64 95)(65 94)(66 93)(67 92)(68 91)(69 90)(70 89)(71 88)(72 87)(73 86)(74 85)(75 84)(76 83)(77 82)(78 108)(79 107)(80 106)(81 105)
G:=sub<Sym(108)| (1,94,29,65)(2,95,30,66)(3,96,31,67)(4,97,32,68)(5,98,33,69)(6,99,34,70)(7,100,35,71)(8,101,36,72)(9,102,37,73)(10,103,38,74)(11,104,39,75)(12,105,40,76)(13,106,41,77)(14,107,42,78)(15,108,43,79)(16,82,44,80)(17,83,45,81)(18,84,46,55)(19,85,47,56)(20,86,48,57)(21,87,49,58)(22,88,50,59)(23,89,51,60)(24,90,52,61)(25,91,53,62)(26,92,54,63)(27,93,28,64), (1,65)(2,66)(3,67)(4,68)(5,69)(6,70)(7,71)(8,72)(9,73)(10,74)(11,75)(12,76)(13,77)(14,78)(15,79)(16,80)(17,81)(18,55)(19,56)(20,57)(21,58)(22,59)(23,60)(24,61)(25,62)(26,63)(27,64)(28,93)(29,94)(30,95)(31,96)(32,97)(33,98)(34,99)(35,100)(36,101)(37,102)(38,103)(39,104)(40,105)(41,106)(42,107)(43,108)(44,82)(45,83)(46,84)(47,85)(48,86)(49,87)(50,88)(51,89)(52,90)(53,91)(54,92), (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), (1,29)(2,28)(3,54)(4,53)(5,52)(6,51)(7,50)(8,49)(9,48)(10,47)(11,46)(12,45)(13,44)(14,43)(15,42)(16,41)(17,40)(18,39)(19,38)(20,37)(21,36)(22,35)(23,34)(24,33)(25,32)(26,31)(27,30)(55,104)(56,103)(57,102)(58,101)(59,100)(60,99)(61,98)(62,97)(63,96)(64,95)(65,94)(66,93)(67,92)(68,91)(69,90)(70,89)(71,88)(72,87)(73,86)(74,85)(75,84)(76,83)(77,82)(78,108)(79,107)(80,106)(81,105)>;
G:=Group( (1,94,29,65)(2,95,30,66)(3,96,31,67)(4,97,32,68)(5,98,33,69)(6,99,34,70)(7,100,35,71)(8,101,36,72)(9,102,37,73)(10,103,38,74)(11,104,39,75)(12,105,40,76)(13,106,41,77)(14,107,42,78)(15,108,43,79)(16,82,44,80)(17,83,45,81)(18,84,46,55)(19,85,47,56)(20,86,48,57)(21,87,49,58)(22,88,50,59)(23,89,51,60)(24,90,52,61)(25,91,53,62)(26,92,54,63)(27,93,28,64), (1,65)(2,66)(3,67)(4,68)(5,69)(6,70)(7,71)(8,72)(9,73)(10,74)(11,75)(12,76)(13,77)(14,78)(15,79)(16,80)(17,81)(18,55)(19,56)(20,57)(21,58)(22,59)(23,60)(24,61)(25,62)(26,63)(27,64)(28,93)(29,94)(30,95)(31,96)(32,97)(33,98)(34,99)(35,100)(36,101)(37,102)(38,103)(39,104)(40,105)(41,106)(42,107)(43,108)(44,82)(45,83)(46,84)(47,85)(48,86)(49,87)(50,88)(51,89)(52,90)(53,91)(54,92), (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), (1,29)(2,28)(3,54)(4,53)(5,52)(6,51)(7,50)(8,49)(9,48)(10,47)(11,46)(12,45)(13,44)(14,43)(15,42)(16,41)(17,40)(18,39)(19,38)(20,37)(21,36)(22,35)(23,34)(24,33)(25,32)(26,31)(27,30)(55,104)(56,103)(57,102)(58,101)(59,100)(60,99)(61,98)(62,97)(63,96)(64,95)(65,94)(66,93)(67,92)(68,91)(69,90)(70,89)(71,88)(72,87)(73,86)(74,85)(75,84)(76,83)(77,82)(78,108)(79,107)(80,106)(81,105) );
G=PermutationGroup([[(1,94,29,65),(2,95,30,66),(3,96,31,67),(4,97,32,68),(5,98,33,69),(6,99,34,70),(7,100,35,71),(8,101,36,72),(9,102,37,73),(10,103,38,74),(11,104,39,75),(12,105,40,76),(13,106,41,77),(14,107,42,78),(15,108,43,79),(16,82,44,80),(17,83,45,81),(18,84,46,55),(19,85,47,56),(20,86,48,57),(21,87,49,58),(22,88,50,59),(23,89,51,60),(24,90,52,61),(25,91,53,62),(26,92,54,63),(27,93,28,64)], [(1,65),(2,66),(3,67),(4,68),(5,69),(6,70),(7,71),(8,72),(9,73),(10,74),(11,75),(12,76),(13,77),(14,78),(15,79),(16,80),(17,81),(18,55),(19,56),(20,57),(21,58),(22,59),(23,60),(24,61),(25,62),(26,63),(27,64),(28,93),(29,94),(30,95),(31,96),(32,97),(33,98),(34,99),(35,100),(36,101),(37,102),(38,103),(39,104),(40,105),(41,106),(42,107),(43,108),(44,82),(45,83),(46,84),(47,85),(48,86),(49,87),(50,88),(51,89),(52,90),(53,91),(54,92)], [(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)], [(1,29),(2,28),(3,54),(4,53),(5,52),(6,51),(7,50),(8,49),(9,48),(10,47),(11,46),(12,45),(13,44),(14,43),(15,42),(16,41),(17,40),(18,39),(19,38),(20,37),(21,36),(22,35),(23,34),(24,33),(25,32),(26,31),(27,30),(55,104),(56,103),(57,102),(58,101),(59,100),(60,99),(61,98),(62,97),(63,96),(64,95),(65,94),(66,93),(67,92),(68,91),(69,90),(70,89),(71,88),(72,87),(73,86),(74,85),(75,84),(76,83),(77,82),(78,108),(79,107),(80,106),(81,105)]])
75 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 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 | 2 | 2 | 2 | 3 | 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 | 27 | 27 | 54 | 54 | 2 | 2 | 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 | D4 | D6 | D6 | D9 | D18 | D18 | D27 | D54 | D54 | S3×D4 | D4×D9 | D4×D27 |
| kernel | D4×D27 | C4×D27 | D108 | C27⋊D4 | D4×C27 | C22×D27 | D4×C9 | D27 | C36 | C2×C18 | C3×D4 | C12 | C2×C6 | D4 | C4 | C22 | C9 | C3 | C1 |
| # reps | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | 3 | 3 | 6 | 9 | 9 | 18 | 1 | 3 | 9 |
Matrix representation of D4×D27 ►in GL4(𝔽109) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 108 | 3 |
| 0 | 0 | 72 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 108 | 3 |
| 0 | 0 | 0 | 1 |
| 87 | 51 | 0 | 0 |
| 58 | 29 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 82 | 59 | 0 | 0 |
| 32 | 27 | 0 | 0 |
| 0 | 0 | 108 | 0 |
| 0 | 0 | 0 | 108 |
G:=sub<GL(4,GF(109))| [1,0,0,0,0,1,0,0,0,0,108,72,0,0,3,1],[1,0,0,0,0,1,0,0,0,0,108,0,0,0,3,1],[87,58,0,0,51,29,0,0,0,0,1,0,0,0,0,1],[82,32,0,0,59,27,0,0,0,0,108,0,0,0,0,108] >;
D4×D27 in GAP, Magma, Sage, TeX
D_4\times D_{27} % in TeX
G:=Group("D4xD27"); // GroupNames label
G:=SmallGroup(432,47);
// by ID
G=gap.SmallGroup(432,47);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,135,2804,557,10085,292,14118]);
// 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,b*d=d*b,d*c*d=c^-1>;
// generators/relations