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G = D24order 48 = 24·3

Dihedral group

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: D24, C31D8, C81S3, C241C2, C4.9D6, C6.2D4, D121C2, C2.4D12, C12.9C22, sometimes denoted D48 or Dih24 or Dih48, SmallGroup(48,7)

Series: Derived Chief Lower central Upper central

C1C12 — D24
C1C3C6C12D12 — D24
C3C6C12 — D24
C1C2C4C8

Generators and relations for D24
 G = < a,b | a24=b2=1, bab=a-1 >

12C2
12C2
6C22
6C22
4S3
4S3
3D4
3D4
2D6
2D6
3D8

Character table of D24

 class 12A2B2C3468A8B12A12B24A24B24C24D
 size 11121222222222222
ρ1111111111111111    trivial
ρ2111-1111-1-111-1-1-1-1    linear of order 2
ρ311-11111-1-111-1-1-1-1    linear of order 2
ρ411-1-111111111111    linear of order 2
ρ52200-12-1-2-2-1-11111    orthogonal lifted from D6
ρ622002-2200-2-20000    orthogonal lifted from D4
ρ72200-12-122-1-1-1-1-1-1    orthogonal lifted from S3
ρ82-20020-2-220022-2-2    orthogonal lifted from D8
ρ92-20020-22-200-2-222    orthogonal lifted from D8
ρ102200-1-2-100113-3-33    orthogonal lifted from D12
ρ112200-1-2-10011-333-3    orthogonal lifted from D12
ρ122-200-1012-2-33ζ87ζ38785ζ3ζ83ζ38ζ38ζ83ζ32838ζ32ζ87ζ3285ζ3285    orthogonal faithful
ρ132-200-101-22-33ζ87ζ3285ζ3285ζ83ζ32838ζ32ζ83ζ38ζ38ζ87ζ38785ζ3    orthogonal faithful
ρ142-200-101-223-3ζ83ζ32838ζ32ζ87ζ3285ζ3285ζ87ζ38785ζ3ζ83ζ38ζ38    orthogonal faithful
ρ152-200-1012-23-3ζ83ζ38ζ38ζ87ζ38785ζ3ζ87ζ3285ζ3285ζ83ζ32838ζ32    orthogonal faithful

Permutation representations of D24
On 24 points - transitive group 24T34
Generators in S24
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24)
(1 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 24)(17 23)(18 22)(19 21)

G:=sub<Sym(24)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,24)(17,23)(18,22)(19,21)>;

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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,24)(17,23)(18,22)(19,21) );

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)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,24),(17,23),(18,22),(19,21)])

G:=TransitiveGroup(24,34);

Matrix representation of D24 in GL2(𝔽23) generated by

1318
57
,
75
1816
G:=sub<GL(2,GF(23))| [13,5,18,7],[7,18,5,16] >;

D24 in GAP, Magma, Sage, TeX

D_{24}
% in TeX

G:=Group("D24");
// GroupNames label

G:=SmallGroup(48,7);
// by ID

G=gap.SmallGroup(48,7);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-3,61,66,182,42,804]);
// Polycyclic

G:=Group<a,b|a^24=b^2=1,b*a*b=a^-1>;
// generators/relations

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