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G = C2×C32⋊2D12  order 432 = 24·33

Direct product of C2 and C32⋊2D12

direct product, non-abelian, soluble, monomial

Aliases: C2×C32⋊2D12, C6⋊1S3≀C2, C3⋊S3⋊2D12, (C3×C6)⋊2D12, C32⋊C4⋊2D6, C33⋊4(C2×D4), (C32×C6)⋊3D4, C32⋊3(C2×D12), C32⋊4D6⋊3C22, C3⋊2(C2×S3≀C2), (C3×C3⋊S3)⋊6D4, (C6×C32⋊C4)⋊5C2, (C2×C32⋊C4)⋊3S3, (C2×C3⋊S3).22D6, (C3×C3⋊S3).7C23, C3⋊S3.4(C22×S3), (C3×C32⋊C4)⋊2C22, (C6×C3⋊S3).35C22, (C2×C32⋊4D6)⋊6C2, SmallGroup(432,756)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32 — C3×C3⋊S3 — C2×C32⋊2D12
Chief series
C1 — C3 — C33 — C3×C3⋊S3 — C32⋊4D6 — C32⋊2D12 — C2×C32⋊2D12
Lower central
C33 — C3×C3⋊S3 — C2×C32⋊2D12
Upper central
C1 — C2

Generators and relations for C2×C32⋊2D12
 G = < a,b,c,d,e | a2=b3=c3=d12=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=c, ebe=dcd-1=b-1, ce=ec, ede=d-1 >

Subgroups: 1616 in 192 conjugacy classes, 31 normal (17 characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C2×C4, D4, C23, C32, C32, C12, D6, C2×C6, C2×D4, C3×S3, C3⋊S3, C3⋊S3, C3×C6, C3×C6, D12, C2×C12, C22×S3, C33, C32⋊C4, S32, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C2×D12, C3×C3⋊S3, C3×C3⋊S3, C32×C6, S3≀C2, C2×C32⋊C4, C2×S32, C3×C32⋊C4, C32⋊4D6, C32⋊4D6, C6×C3⋊S3, C6×C3⋊S3, C2×S3≀C2, C32⋊2D12, C6×C32⋊C4, C2×C32⋊4D6, C2×C32⋊2D12
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D12, C22×S3, C2×D12, S3≀C2, C2×S3≀C2, C32⋊2D12, C2×C32⋊2D12

Character table of C2×C32⋊2D12

 class 12A2B2C2D2E2F2G3A3B3C3D3E4A4B6A6B6C6D6E6F6G6H6I6J6K12A12B12C12D
 size 1199181818182448818182448818183636363618181818
ρ1111111111111111111111111111111    trivial
ρ21111-1-1-1-111111111111111-1-1-1-11111    linear of order 2
ρ31-11-11-11-1111111-1-1-1-1-1-1-111-11-1-11-11    linear of order 2
ρ41-11-1-11-11111111-1-1-1-1-1-1-11-11-11-11-11    linear of order 2
ρ51-11-111-1-111111-11-1-1-1-1-1-1111-1-11-11-1    linear of order 2
ρ61-11-1-1-11111111-11-1-1-1-1-1-11-1-1111-11-1    linear of order 2
ρ711111-1-1111111-1-111111111-1-11-1-1-1-1    linear of order 2
ρ81111-111-111111-1-11111111-111-1-1-1-1-1    linear of order 2
ρ92-22-20000-122-1-1-221-2-2111-10000-11-11    orthogonal lifted from D6
ρ102-2-2200002222200-2-2-2-2-22-200000000    orthogonal lifted from D4
ρ1122220000-122-1-122-122-1-1-1-10000-1-1-1-1    orthogonal lifted from S3
ρ122-22-20000-122-1-12-21-2-2111-100001-11-1    orthogonal lifted from D6
ρ1322220000-122-1-1-2-2-122-1-1-1-100001111    orthogonal lifted from D6
ρ1422-2-20000222220022222-2-200000000    orthogonal lifted from D4
ρ1522-2-20000-122-1-100-122-1-1110000√3√3-√3-√3    orthogonal lifted from D12
ρ1622-2-20000-122-1-100-122-1-1110000-√3-√3√3√3    orthogonal lifted from D12
ρ172-2-220000-122-1-1001-2-211-110000-√3√3√3-√3    orthogonal lifted from D12
ρ182-2-220000-122-1-1001-2-211-110000√3-√3-√3√3    orthogonal lifted from D12
ρ194-400200-241-2-2100-4-12-1200-10010000    orthogonal lifted from C2×S3≀C2
ρ204-4000-2204-211-200-42-12-10001-100000    orthogonal lifted from C2×S3≀C2
ρ2144000-2-204-211-2004-21-210001100000    orthogonal lifted from S3≀C2
ρ224400200241-2-210041-21-200-100-10000    orthogonal lifted from S3≀C2
ρ234-40002-204-211-200-42-12-1000-1100000    orthogonal lifted from C2×S3≀C2
ρ24440002204-211-2004-21-21000-1-100000    orthogonal lifted from S3≀C2
ρ254400-200-241-2-210041-21-20010010000    orthogonal lifted from S3≀C2
ρ264-400-200241-2-2100-4-12-1200100-10000    orthogonal lifted from C2×S3≀C2
ρ278-8000000-4-42-120044-2-210000000000    orthogonal faithful
ρ2888000000-42-42-100-42-4-120000000000    orthogonal lifted from C32⋊2D12
ρ298-8000000-42-42-1004-241-20000000000    orthogonal faithful
ρ3088000000-4-42-1200-4-422-10000000000    orthogonal lifted from C32⋊2D12

Permutation representations of C2×C32⋊2D12
►On 24 points - transitive group 24T1304
Generators in S24
(1 24)(2 13)(3 14)(4 15)(5 16)(6 17)(7 18)(8 19)(9 20)(10 21)(11 22)(12 23)

(1 9 5)(2 6 10)(3 7 11)(4 12 8)(13 17 21)(14 18 22)(15 23 19)(16 24 20)

(1 5 9)(2 6 10)(3 11 7)(4 12 8)(13 17 21)(14 22 18)(15 23 19)(16 20 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 23)(2 22)(3 21)(4 20)(5 19)(6 18)(7 17)(8 16)(9 15)(10 14)(11 13)(12 24)
 
G:=sub<Sym(24)| (1,24)(2,13)(3,14)(4,15)(5,16)(6,17)(7,18)(8,19)(9,20)(10,21)(11,22)(12,23), (1,9,5)(2,6,10)(3,7,11)(4,12,8)(13,17,21)(14,18,22)(15,23,19)(16,24,20), (1,5,9)(2,6,10)(3,11,7)(4,12,8)(13,17,21)(14,22,18)(15,23,19)(16,20,24), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,23)(2,22)(3,21)(4,20)(5,19)(6,18)(7,17)(8,16)(9,15)(10,14)(11,13)(12,24)>;
 
G:=Group( (1,24)(2,13)(3,14)(4,15)(5,16)(6,17)(7,18)(8,19)(9,20)(10,21)(11,22)(12,23), (1,9,5)(2,6,10)(3,7,11)(4,12,8)(13,17,21)(14,18,22)(15,23,19)(16,24,20), (1,5,9)(2,6,10)(3,11,7)(4,12,8)(13,17,21)(14,22,18)(15,23,19)(16,20,24), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,23)(2,22)(3,21)(4,20)(5,19)(6,18)(7,17)(8,16)(9,15)(10,14)(11,13)(12,24) );
 
G=PermutationGroup([[(1,24),(2,13),(3,14),(4,15),(5,16),(6,17),(7,18),(8,19),(9,20),(10,21),(11,22),(12,23)], [(1,9,5),(2,6,10),(3,7,11),(4,12,8),(13,17,21),(14,18,22),(15,23,19),(16,24,20)], [(1,5,9),(2,6,10),(3,11,7),(4,12,8),(13,17,21),(14,22,18),(15,23,19),(16,20,24)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24)], [(1,23),(2,22),(3,21),(4,20),(5,19),(6,18),(7,17),(8,16),(9,15),(10,14),(11,13),(12,24)]])
 
G:=TransitiveGroup(24,1304);
 

Matrix representation of C2×C32⋊2D12 ►in GL6(𝔽13)

1200000
0120000
0012000
0001200
0000120
0000012
,
100000
010000
0012001
000100
000010
0012000
,
100000
010000
001000
0001210
0001200
000001
,
3100000
360000
000100
000001
001000
000010
,
1030000
630000
0000012
0001200
0000120
0012000

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,12,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,1,0,0,0,0,0,0,0,1],[3,3,0,0,0,0,10,6,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0],[10,6,0,0,0,0,3,3,0,0,0,0,0,0,0,0,0,12,0,0,0,12,0,0,0,0,0,0,12,0,0,0,12,0,0,0] >;
 

C2×C32⋊2D12 in GAP, Magma, Sage, TeX 

C_2\times C_3^2\rtimes_2D_{12}
 
% in TeX
 
G:=Group("C2xC3^2:2D12");
 
// GroupNames label
 
G:=SmallGroup(432,756);
 
// by ID
 
G=gap.SmallGroup(432,756);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,141,64,1684,1691,165,677,348,530,14118]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^3=c^3=d^12=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=c,e*b*e=d*c*d^-1=b^-1,c*e=e*c,e*d*e=d^-1>;
 
// generators/relations
 

Export

Character table of C2×C32⋊2D12 in TeX

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