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G = D56order 112 = 24·7

Dihedral group

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: D56, C71D8, C81D7, C561C2, D281C2, C2.4D28, C4.9D14, C14.2D4, C28.9C22, sometimes denoted D112 or Dih56 or Dih112, SmallGroup(112,6)

Series: Derived Chief Lower central Upper central

C1C28 — D56
C1C7C14C28D28 — D56
C7C14C28 — D56
C1C2C4C8

Generators and relations for D56
 G = < a,b | a56=b2=1, bab=a-1 >

28C2
28C2
14C22
14C22
4D7
4D7
7D4
7D4
2D14
2D14
7D8

Smallest permutation representation of D56
On 56 points
Generators in S56
(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)
(1 56)(2 55)(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)(28 29)

G:=sub<Sym(56)| (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), (1,56)(2,55)(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)(28,29)>;

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), (1,56)(2,55)(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)(28,29) );

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)], [(1,56),(2,55),(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),(28,29)])

D56 is a maximal subgroup of
D112  C112⋊C2  C7⋊D16  C7⋊SD32  D567C2  C8⋊D14  D7×D8  D56⋊C2  Q8.D14  D56⋊C3  C3⋊D56  D168
D56 is a maximal quotient of
D112  C112⋊C2  Dic56  C561C4  C2.D56  C3⋊D56  D168

31 conjugacy classes

class 1 2A2B2C 4 7A7B7C8A8B14A14B14C28A···28F56A···56L
order122247778814141428···2856···56
size1128282222222222···22···2

31 irreducible representations

dim111222222
type+++++++++
imageC1C2C2D4D7D8D14D28D56
kernelD56C56D28C14C8C7C4C2C1
# reps1121323612

Matrix representation of D56 in GL2(𝔽113) generated by

4944
2534
,
21106
7992
G:=sub<GL(2,GF(113))| [49,25,44,34],[21,79,106,92] >;

D56 in GAP, Magma, Sage, TeX

D_{56}
% in TeX

G:=Group("D56");
// GroupNames label

G:=SmallGroup(112,6);
// by ID

G=gap.SmallGroup(112,6);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-7,61,66,182,42,2404]);
// Polycyclic

G:=Group<a,b|a^56=b^2=1,b*a*b=a^-1>;
// generators/relations

Export

Subgroup lattice of D56 in TeX

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