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G = C32⋊2D24  order 432 = 24·33

The semidirect product of C32 and D24 acting via D24/C6=D4

non-abelian, soluble, monomial

Aliases: C33⋊3D8, C32⋊2D24, C6.9S3≀C2, C3⋊1(C32⋊D8), C32⋊2C8⋊1S3, (C3×C6).10D12, C33⋊9D4⋊1C2, C3⋊Dic3.14D6, (C32×C6).15D4, C2.3(C32⋊2D12), (C3×C32⋊2C8)⋊1C2, (C3×C3⋊Dic3).1C22, SmallGroup(432,588)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32 — C3×C3⋊Dic3 — C32⋊2D24
Chief series
C1 — C3 — C33 — C32×C6 — C3×C3⋊Dic3 — C33⋊9D4 — C32⋊2D24
Lower central
C33 — C32×C6 — C3×C3⋊Dic3 — C32⋊2D24
Upper central
C1 — C2

Generators and relations for C32⋊2D24
 G = < a,b,c,d | a3=b3=c24=d2=1, ab=ba, cac-1=b, dad=cbc-1=a-1, bd=db, dcd=c-1 >

Subgroups: 816 in 96 conjugacy classes, 15 normal (13 characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C8, D4, C32, C32, Dic3, C12, D6, C2×C6, D8, C3×S3, C3⋊S3, C3×C6, C3×C6, C24, D12, C3⋊D4, C33, C3×Dic3, C3⋊Dic3, S3×C6, C2×C3⋊S3, D24, C3×C3⋊S3, C32×C6, C32⋊2C8, D6⋊S3, C3⋊D12, C3×C3⋊Dic3, C6×C3⋊S3, C32⋊D8, C3×C32⋊2C8, C33⋊9D4, C32⋊2D24
Quotients: C1, C2, C22, S3, D4, D6, D8, D12, D24, S3≀C2, C32⋊D8, C32⋊2D12, C32⋊2D24

Character table of C32⋊2D24

 class 12A2B2C3A3B3C3D3E46A6B6C6D6E6F6G6H6I8A8B12A12B24A24B24C24D
 size 113636244881824488363636361818181818181818
ρ1111111111111111111111111111    trivial
ρ211-1111111111111-11-11-1-111-1-1-1-1    linear of order 2
ρ311-1-111111111111-1-1-1-111111111    linear of order 2
ρ4111-1111111111111-11-1-1-111-1-1-1-1    linear of order 2
ρ52200-122-1-12-122-1-10000-2-2-1-11111    orthogonal lifted from D6
ρ6220022222-222222000000-2-20000    orthogonal lifted from D4
ρ72200-122-1-12-122-1-1000022-1-1-1-1-1-1    orthogonal lifted from S3
ρ82-200222220-2-2-2-2-20000-√2√200-√2√2√2-√2    orthogonal lifted from D8
ρ92-200222220-2-2-2-2-20000√2-√200√2-√2-√2√2    orthogonal lifted from D8
ρ102200-122-1-1-2-122-1-100000011-√3-√3√3√3    orthogonal lifted from D12
ρ112200-122-1-1-2-122-1-100000011√3√3-√3-√3    orthogonal lifted from D12
ρ122-200-122-1-101-2-2110000√2-√2√3-√3ζ83ζ3+ζ83+ζ8ζ3ζ83ζ32+ζ8ζ32+ζ8ζ87ζ32+ζ87+ζ85ζ32ζ87ζ3+ζ85ζ3+ζ85    orthogonal lifted from D24
ρ132-200-122-1-101-2-2110000√2-√2-√3√3ζ87ζ3+ζ85ζ3+ζ85ζ87ζ32+ζ87+ζ85ζ32ζ83ζ32+ζ8ζ32+ζ8ζ83ζ3+ζ83+ζ8ζ3    orthogonal lifted from D24
ρ142-200-122-1-101-2-2110000-√2√2√3-√3ζ83ζ32+ζ8ζ32+ζ8ζ83ζ3+ζ83+ζ8ζ3ζ87ζ3+ζ85ζ3+ζ85ζ87ζ32+ζ87+ζ85ζ32    orthogonal lifted from D24
ρ152-200-122-1-101-2-2110000-√2√2-√3√3ζ87ζ32+ζ87+ζ85ζ32ζ87ζ3+ζ85ζ3+ζ85ζ83ζ3+ζ83+ζ8ζ3ζ83ζ32+ζ8ζ32+ζ8    orthogonal lifted from D24
ρ1644-204-211-2041-2-21101000000000    orthogonal lifted from S3≀C2
ρ17440-241-2-2104-211-2010100000000    orthogonal lifted from S3≀C2
ρ18440241-2-2104-211-20-10-100000000    orthogonal lifted from S3≀C2
ρ1944204-211-2041-2-21-10-1000000000    orthogonal lifted from S3≀C2
ρ204-4004-211-20-4-122-1√-30-√-3000000000    complex lifted from C32⋊D8
ρ214-40041-2-210-42-1-120-√-30√-300000000    complex lifted from C32⋊D8
ρ224-4004-211-20-4-122-1-√-30√-3000000000    complex lifted from C32⋊D8
ρ234-40041-2-210-42-1-120√-30-√-300000000    complex lifted from C32⋊D8
ρ248-800-42-42-1044-21-2000000000000    orthogonal faithful
ρ258800-4-42-120-42-42-1000000000000    orthogonal lifted from C32⋊2D12
ρ268800-42-42-10-4-42-12000000000000    orthogonal lifted from C32⋊2D12
ρ278-800-4-42-1204-24-21000000000000    orthogonal faithful

Permutation representations of C32⋊2D24
►On 24 points - transitive group 24T1306
Generators in S24
(1 9 17)(2 18 10)(3 19 11)(4 12 20)(5 13 21)(6 22 14)(7 23 15)(8 16 24)

(1 17 9)(2 18 10)(3 11 19)(4 12 20)(5 21 13)(6 22 14)(7 15 23)(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 24)(2 23)(3 22)(4 21)(5 20)(6 19)(7 18)(8 17)(9 16)(10 15)(11 14)(12 13)
 
G:=sub<Sym(24)| (1,9,17)(2,18,10)(3,19,11)(4,12,20)(5,13,21)(6,22,14)(7,23,15)(8,16,24), (1,17,9)(2,18,10)(3,11,19)(4,12,20)(5,21,13)(6,22,14)(7,15,23)(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), (1,24)(2,23)(3,22)(4,21)(5,20)(6,19)(7,18)(8,17)(9,16)(10,15)(11,14)(12,13)>;
 
G:=Group( (1,9,17)(2,18,10)(3,19,11)(4,12,20)(5,13,21)(6,22,14)(7,23,15)(8,16,24), (1,17,9)(2,18,10)(3,11,19)(4,12,20)(5,21,13)(6,22,14)(7,15,23)(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), (1,24)(2,23)(3,22)(4,21)(5,20)(6,19)(7,18)(8,17)(9,16)(10,15)(11,14)(12,13) );
 
G=PermutationGroup([[(1,9,17),(2,18,10),(3,19,11),(4,12,20),(5,13,21),(6,22,14),(7,23,15),(8,16,24)], [(1,17,9),(2,18,10),(3,11,19),(4,12,20),(5,21,13),(6,22,14),(7,15,23),(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)], [(1,24),(2,23),(3,22),(4,21),(5,20),(6,19),(7,18),(8,17),(9,16),(10,15),(11,14),(12,13)]])
 
G:=TransitiveGroup(24,1306);
 

Matrix representation of C32⋊2D24 ►in GL6(𝔽73)

100000
010000
00720720
000100
001000
000001
,
100000
010000
001000
00072072
000010
000100
,
68230000
50180000
000100
001000
000001
00720720
,
50180000
68230000
001000
000100
00720720
000001

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,1,0,0,0,0,1,0,0,0,0,72,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,1,0,0,0,0,1,0,0,0,0,72,0,0],[68,50,0,0,0,0,23,18,0,0,0,0,0,0,0,1,0,72,0,0,1,0,0,0,0,0,0,0,0,72,0,0,0,0,1,0],[50,68,0,0,0,0,18,23,0,0,0,0,0,0,1,0,72,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,1] >;
 

C32⋊2D24 in GAP, Magma, Sage, TeX 

C_3^2\rtimes_2D_{24}
 
% in TeX
 
G:=Group("C3^2:2D24");
 
// GroupNames label
 
G:=SmallGroup(432,588);
 
// by ID
 
G=gap.SmallGroup(432,588);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,85,92,254,58,1684,1691,298,677,348,1027,14118]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^3=b^3=c^24=d^2=1,a*b=b*a,c*a*c^-1=b,d*a*d=c*b*c^-1=a^-1,b*d=d*b,d*c*d=c^-1>;
 
// generators/relations
 

Export

Character table of C32⋊2D24 in TeX

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