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G = C3⋊D24  order 144 = 24·32

The semidirect product of C3 and D24 acting via D24/D12=C2

metabelian, supersoluble, monomial

Aliases: C3⋊2D24, C32⋊3D8, D12⋊1S3, C12.10D6, C6.12D12, C3⋊C8⋊1S3, C4.1S32, (C3×C6).7D4, C3⋊1(D4⋊S3), (C3×D12)⋊2C2, C12⋊S3⋊2C2, C6.1(C3⋊D4), (C3×C12).2C22, C2.4(C3⋊D12), (C3×C3⋊C8)⋊1C2, SmallGroup(144,57)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C12 — C3⋊D24
Chief series
C1 — C3 — C32 — C3×C6 — C3×C12 — C3×D12 — C3⋊D24
Lower central
C32 — C3×C6 — C3×C12 — C3⋊D24
Upper central
C1 — C2 — C4

Generators and relations for C3⋊D24
 G = < a,b,c | a3=b24=c2=1, bab-1=cac=a-1, cbc=b-1 >

C1
C2
12C2
36C2
C3
C3
2C3
C4
6C22
18C22
C6
C6
2C6
4S3
12S3
12S3
12S3
12C6
12S3
C32
3C8
3D4
9D4
C12
C12
2C12
2D6
6D6
6D6
6D6
6D6
6C2×C6
C3×C6
4C3×S3
4C3⋊S3
9D8
C3⋊C8
D12
3C24
3D12
3C3×D4
3D12
6D12
C3×C12
2S3×C6
2C2×C3⋊S3
3D24
3D4⋊S3
C3×D12
C12⋊S3
C3×C3⋊C8
C3⋊D24

Character table of C3⋊D24

 class 12A2B2C3A3B3C46A6B6C6D6E8A8B12A12B12C12D12E24A24B24C24D
 size 1112362242224121266224446666
ρ1111111111111111111111111    trivial
ρ211-111111111-1-1-1-111111-1-1-1-1    linear of order 2
ρ311-1-11111111-1-111111111111    linear of order 2
ρ4111-1111111111-1-111111-1-1-1-1    linear of order 2
ρ52200-12-122-1-10022-1-1-1-12-1-1-1-1    orthogonal lifted from S3
ρ622202-1-12-12-1-1-10022-1-1-10000    orthogonal lifted from S3
ρ722-202-1-12-12-1110022-1-1-10000    orthogonal lifted from D6
ρ82200-12-122-1-100-2-2-1-1-1-121111    orthogonal lifted from D6
ρ92200222-22220000-2-2-2-2-20000    orthogonal lifted from D4
ρ102200-12-1-22-1-100001111-2-√3-√3√3√3    orthogonal lifted from D12
ρ112200-12-1-22-1-100001111-2√3√3-√3-√3    orthogonal lifted from D12
ρ122-2002220-2-2-200√2-√200000√2-√2-√2√2    orthogonal lifted from D8
ρ132-2002220-2-2-200-√2√200000-√2√2√2-√2    orthogonal lifted from D8
ρ142-200-12-10-21100√2-√2√3-√3√3-√30ζ87ζ3+ζ85ζ3+ζ85ζ87ζ32+ζ87+ζ85ζ32ζ83ζ32+ζ8ζ32+ζ8ζ83ζ3+ζ83+ζ8ζ3    orthogonal lifted from D24
ρ152-200-12-10-21100-√2√2-√3√3-√3√30ζ83ζ32+ζ8ζ32+ζ8ζ83ζ3+ζ83+ζ8ζ3ζ87ζ3+ζ85ζ3+ζ85ζ87ζ32+ζ87+ζ85ζ32    orthogonal lifted from D24
ρ162-200-12-10-21100√2-√2-√3√3-√3√30ζ83ζ3+ζ83+ζ8ζ3ζ83ζ32+ζ8ζ32+ζ8ζ87ζ32+ζ87+ζ85ζ32ζ87ζ3+ζ85ζ3+ζ85    orthogonal lifted from D24
ρ172-200-12-10-21100-√2√2√3-√3√3-√30ζ87ζ32+ζ87+ζ85ζ32ζ87ζ3+ζ85ζ3+ζ85ζ83ζ3+ζ83+ζ8ζ3ζ83ζ32+ζ8ζ32+ζ8    orthogonal lifted from D24
ρ1822002-1-1-2-12-1√-3-√-300-2-21110000    complex lifted from C3⋊D4
ρ1922002-1-1-2-12-1-√-3√-300-2-21110000    complex lifted from C3⋊D4
ρ204-4004-2-202-420000000000000    orthogonal lifted from D4⋊S3, Schur index 2
ρ214400-2-214-2-210000-2-211-20000    orthogonal lifted from S32
ρ224400-2-21-4-2-21000022-1-120000    orthogonal lifted from C3⋊D12
ρ234-400-2-21022-10000-2√32√3√3-√300000    orthogonal faithful
ρ244-400-2-21022-100002√3-2√3-√3√300000    orthogonal faithful

Permutation representations of C3⋊D24
►On 24 points - transitive group 24T234
Generators in S24
(1 17 9)(2 10 18)(3 19 11)(4 12 20)(5 21 13)(6 14 22)(7 23 15)(8 16 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:=sub<Sym(24)| (1,17,9)(2,10,18)(3,19,11)(4,12,20)(5,21,13)(6,14,22)(7,23,15)(8,16,24), (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( (1,17,9)(2,10,18)(3,19,11)(4,12,20)(5,21,13)(6,14,22)(7,23,15)(8,16,24), (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=PermutationGroup([[(1,17,9),(2,10,18),(3,19,11),(4,12,20),(5,21,13),(6,14,22),(7,23,15),(8,16,24)], [(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:=TransitiveGroup(24,234);
 

C3⋊D24 is a maximal subgroup of
 S3×D24  C24⋊1D6  D24⋊S3  D6.1D12  D12⋊18D6  D12.27D6  D12.28D6  S3×D4⋊S3  D12⋊D6  D12.7D6  D12⋊5D6  D12⋊6D6  D12.10D6  D12.13D6  D12.14D6  C3⋊D72  C9⋊D24  He3⋊2D8  He3⋊3D8  C33⋊7D8  C33⋊8D8  C33⋊9D8
C3⋊D24 is a maximal quotient of
 C3⋊D48  C32⋊3SD32  C24.49D6  C32⋊3Q32  C6.16D24  C6.17D24  C6.18D24  C3⋊D72  C9⋊D24  He3⋊3D8  C33⋊7D8  C33⋊8D8  C33⋊9D8

Matrix representation of C3⋊D24 ►in GL6(𝔽73)

100000
010000
001000
000100
0000072
0000172
,
13480000
3280000
001100
0072000
000001
000010
,
100000
14720000
00727200
000100
000001
000010

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[13,3,0,0,0,0,48,28,0,0,0,0,0,0,1,72,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,14,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,72,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
 

C3⋊D24 in GAP, Magma, Sage, TeX 

C_3\rtimes D_{24}
 
% in TeX
 
G:=Group("C3:D24");
 
// GroupNames label
 
G:=SmallGroup(144,57);
 
// by ID
 
G=gap.SmallGroup(144,57);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-2,-3,-3,73,79,218,50,490,3461]);
 
// Polycyclic
 
G:=Group<a,b,c|a^3=b^24=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
 
// generators/relations
 

Export

Subgroup lattice of C3⋊D24 in TeX
Character table of C3⋊D24 in TeX

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