direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: D4×C27, C4⋊C54, C108⋊3C2, C36.7C6, C12.3C18, C22⋊2C54, C54.6C22, C9.(C3×D4), C3.(D4×C9), C54○(D4×C9), (D4×C9).C3, (C3×D4).C9, (C2×C54)⋊1C2, (C2×C6).2C18, C6.6(C2×C18), (C2×C18).4C6, C2.1(C2×C54), C18.14(C2×C6), SmallGroup(216,10)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4×C27
G = < a,b,c | a27=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of D4×C27
►On 108 points
Generators in S108
(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 53 65 82)(2 54 66 83)(3 28 67 84)(4 29 68 85)(5 30 69 86)(6 31 70 87)(7 32 71 88)(8 33 72 89)(9 34 73 90)(10 35 74 91)(11 36 75 92)(12 37 76 93)(13 38 77 94)(14 39 78 95)(15 40 79 96)(16 41 80 97)(17 42 81 98)(18 43 55 99)(19 44 56 100)(20 45 57 101)(21 46 58 102)(22 47 59 103)(23 48 60 104)(24 49 61 105)(25 50 62 106)(26 51 63 107)(27 52 64 108)
(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)
G:=sub<Sym(108)| (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), (1,53,65,82)(2,54,66,83)(3,28,67,84)(4,29,68,85)(5,30,69,86)(6,31,70,87)(7,32,71,88)(8,33,72,89)(9,34,73,90)(10,35,74,91)(11,36,75,92)(12,37,76,93)(13,38,77,94)(14,39,78,95)(15,40,79,96)(16,41,80,97)(17,42,81,98)(18,43,55,99)(19,44,56,100)(20,45,57,101)(21,46,58,102)(22,47,59,103)(23,48,60,104)(24,49,61,105)(25,50,62,106)(26,51,63,107)(27,52,64,108), (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)>;
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), (1,53,65,82)(2,54,66,83)(3,28,67,84)(4,29,68,85)(5,30,69,86)(6,31,70,87)(7,32,71,88)(8,33,72,89)(9,34,73,90)(10,35,74,91)(11,36,75,92)(12,37,76,93)(13,38,77,94)(14,39,78,95)(15,40,79,96)(16,41,80,97)(17,42,81,98)(18,43,55,99)(19,44,56,100)(20,45,57,101)(21,46,58,102)(22,47,59,103)(23,48,60,104)(24,49,61,105)(25,50,62,106)(26,51,63,107)(27,52,64,108), (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) );
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)], [(1,53,65,82),(2,54,66,83),(3,28,67,84),(4,29,68,85),(5,30,69,86),(6,31,70,87),(7,32,71,88),(8,33,72,89),(9,34,73,90),(10,35,74,91),(11,36,75,92),(12,37,76,93),(13,38,77,94),(14,39,78,95),(15,40,79,96),(16,41,80,97),(17,42,81,98),(18,43,55,99),(19,44,56,100),(20,45,57,101),(21,46,58,102),(22,47,59,103),(23,48,60,104),(24,49,61,105),(25,50,62,106),(26,51,63,107),(27,52,64,108)], [(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)]])
D4×C27 is a maximal subgroup of
D4.D27 D4⋊D27 D4⋊2D27
135 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4 | 6A | 6B | 6C | 6D | 6E | 6F | 9A | ··· | 9F | 12A | 12B | 18A | ··· | 18F | 18G | ··· | 18R | 27A | ··· | 27R | 36A | ··· | 36F | 54A | ··· | 54R | 54S | ··· | 54BB | 108A | ··· | 108R |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 9 | ··· | 9 | 12 | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 27 | ··· | 27 | 36 | ··· | 36 | 54 | ··· | 54 | 54 | ··· | 54 | 108 | ··· | 108 |
| size | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
135 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | C9 | C18 | C18 | C27 | C54 | C54 | D4 | C3×D4 | D4×C9 | D4×C27 |
| kernel | D4×C27 | C108 | C2×C54 | D4×C9 | C36 | C2×C18 | C3×D4 | C12 | C2×C6 | D4 | C4 | C22 | C27 | C9 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 6 | 6 | 12 | 18 | 18 | 36 | 1 | 2 | 6 | 18 |
Matrix representation of D4×C27 ►in GL2(𝔽109) generated by
| 21 | 0 |
| 0 | 21 |
| 1 | 62 |
| 7 | 108 |
| 1 | 62 |
| 0 | 108 |
G:=sub<GL(2,GF(109))| [21,0,0,21],[1,7,62,108],[1,0,62,108] >;
D4×C27 in GAP, Magma, Sage, TeX
D_4\times C_{27} % in TeX
G:=Group("D4xC27"); // GroupNames label
G:=SmallGroup(216,10);
// by ID
G=gap.SmallGroup(216,10);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-3,-3,169,122,118]);
// Polycyclic
G:=Group<a,b,c|a^27=b^4=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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