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G = D108order 216 = 23·33

Dihedral group

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: D108, C4⋊D27, C271D4, C3.D36, C9.D12, C1081C2, D541C2, C36.2S3, C12.2D9, C2.4D54, C6.10D18, C18.10D6, C54.3C22, sometimes denoted D216 or Dih108 or Dih216, SmallGroup(216,6)

Series: Derived Chief Lower central Upper central

C1C54 — D108
C1C3C9C27C54D54 — D108
C27C54 — D108
C1C2C4

Generators and relations for D108
 G = < a,b | a108=b2=1, bab=a-1 >

54C2
54C2
27C22
27C22
18S3
18S3
27D4
9D6
9D6
6D9
6D9
9D12
3D18
3D18
2D27
2D27
3D36

Smallest permutation representation of D108
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)
(1 108)(2 107)(3 106)(4 105)(5 104)(6 103)(7 102)(8 101)(9 100)(10 99)(11 98)(12 97)(13 96)(14 95)(15 94)(16 93)(17 92)(18 91)(19 90)(20 89)(21 88)(22 87)(23 86)(24 85)(25 84)(26 83)(27 82)(28 81)(29 80)(30 79)(31 78)(32 77)(33 76)(34 75)(35 74)(36 73)(37 72)(38 71)(39 70)(40 69)(41 68)(42 67)(43 66)(44 65)(45 64)(46 63)(47 62)(48 61)(49 60)(50 59)(51 58)(52 57)(53 56)(54 55)

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), (1,108)(2,107)(3,106)(4,105)(5,104)(6,103)(7,102)(8,101)(9,100)(10,99)(11,98)(12,97)(13,96)(14,95)(15,94)(16,93)(17,92)(18,91)(19,90)(20,89)(21,88)(22,87)(23,86)(24,85)(25,84)(26,83)(27,82)(28,81)(29,80)(30,79)(31,78)(32,77)(33,76)(34,75)(35,74)(36,73)(37,72)(38,71)(39,70)(40,69)(41,68)(42,67)(43,66)(44,65)(45,64)(46,63)(47,62)(48,61)(49,60)(50,59)(51,58)(52,57)(53,56)(54,55)>;

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), (1,108)(2,107)(3,106)(4,105)(5,104)(6,103)(7,102)(8,101)(9,100)(10,99)(11,98)(12,97)(13,96)(14,95)(15,94)(16,93)(17,92)(18,91)(19,90)(20,89)(21,88)(22,87)(23,86)(24,85)(25,84)(26,83)(27,82)(28,81)(29,80)(30,79)(31,78)(32,77)(33,76)(34,75)(35,74)(36,73)(37,72)(38,71)(39,70)(40,69)(41,68)(42,67)(43,66)(44,65)(45,64)(46,63)(47,62)(48,61)(49,60)(50,59)(51,58)(52,57)(53,56)(54,55) );

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)], [(1,108),(2,107),(3,106),(4,105),(5,104),(6,103),(7,102),(8,101),(9,100),(10,99),(11,98),(12,97),(13,96),(14,95),(15,94),(16,93),(17,92),(18,91),(19,90),(20,89),(21,88),(22,87),(23,86),(24,85),(25,84),(26,83),(27,82),(28,81),(29,80),(30,79),(31,78),(32,77),(33,76),(34,75),(35,74),(36,73),(37,72),(38,71),(39,70),(40,69),(41,68),(42,67),(43,66),(44,65),(45,64),(46,63),(47,62),(48,61),(49,60),(50,59),(51,58),(52,57),(53,56),(54,55)])

D108 is a maximal subgroup of   C216⋊C2  D216  D4⋊D27  Q82D27  D1085C2  D4×D27  Q83D27
D108 is a maximal quotient of   Dic108  C216⋊C2  D216  C4⋊Dic27  D54⋊C4

57 conjugacy classes

class 1 2A2B2C 3  4  6 9A9B9C12A12B18A18B18C27A···27I36A···36F54A···54I108A···108R
order1222346999121218181827···2736···3654···54108···108
size115454222222222222···22···22···22···2

57 irreducible representations

dim1112222222222
type+++++++++++++
imageC1C2C2S3D4D6D9D12D18D27D36D54D108
kernelD108C108D54C36C27C18C12C9C6C4C3C2C1
# reps11211132396918

Matrix representation of D108 in GL2(𝔽109) generated by

7866
4335
,
4335
7866
G:=sub<GL(2,GF(109))| [78,43,66,35],[43,78,35,66] >;

D108 in GAP, Magma, Sage, TeX

D_{108}
% in TeX

G:=Group("D108");
// GroupNames label

G:=SmallGroup(216,6);
// by ID

G=gap.SmallGroup(216,6);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-3,73,31,963,381,3604,208,5189]);
// Polycyclic

G:=Group<a,b|a^108=b^2=1,b*a*b=a^-1>;
// generators/relations

Export

Subgroup lattice of D108 in TeX

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