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G = D6⋊(C32⋊C4)  order 432 = 24·33

The semidirect product of D6 and C32⋊C4 acting via C32⋊C4/C3⋊S3=C2

metabelian, soluble, monomial

Aliases: D6⋊(C32⋊C4), C3⋊S3.5D12, C32⋊7(D6⋊C4), C33⋊1(C22⋊C4), C3⋊1(C62⋊C4), (S3×C3×C6)⋊1C4, (C6×C32⋊C4)⋊1C2, (C2×C32⋊C4)⋊1S3, (C3×C3⋊S3).7D4, C6.2(C2×C32⋊C4), C2.4(S3×C32⋊C4), (C3×C6).27(C4×S3), (C2×C3⋊S3).25D6, (C2×C33⋊C2)⋊1C4, (C2×C33⋊C4)⋊1C2, C3⋊S3.7(C3⋊D4), (C6×C3⋊S3).2C22, (C32×C6).2(C2×C4), (C2×S3×C3⋊S3).1C2, SmallGroup(432,568)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32×C6 — D6⋊(C32⋊C4)
Chief series
C1 — C3 — C33 — C3×C3⋊S3 — C6×C3⋊S3 — C2×S3×C3⋊S3 — D6⋊(C32⋊C4)
Lower central
C33 — C32×C6 — D6⋊(C32⋊C4)
Upper central
C1 — C2

Generators and relations for D6⋊(C32⋊C4)
 G = < a,b,c,d,e | a6=b2=c3=d3=e4=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=a3b, ede-1=cd=dc, ece-1=c-1d >

Subgroups: 1472 in 152 conjugacy classes, 22 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C2×C4, C23, C32, C32, Dic3, C12, D6, D6, C2×C6, C22⋊C4, C3×S3, C3⋊S3, C3⋊S3, C3×C6, C3×C6, C2×Dic3, C2×C12, C22×S3, C33, C32⋊C4, S32, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C62, D6⋊C4, S3×C32, C3×C3⋊S3, C33⋊C2, C32×C6, C2×C32⋊C4, C2×C32⋊C4, C2×S32, C22×C3⋊S3, C3×C32⋊C4, C33⋊C4, S3×C3⋊S3, S3×C3×C6, C6×C3⋊S3, C2×C33⋊C2, C62⋊C4, C6×C32⋊C4, C2×C33⋊C4, C2×S3×C3⋊S3, D6⋊(C32⋊C4)
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D6, C22⋊C4, C4×S3, D12, C3⋊D4, C32⋊C4, D6⋊C4, C2×C32⋊C4, C62⋊C4, S3×C32⋊C4, D6⋊(C32⋊C4)

Character table of D6⋊(C32⋊C4)

 class 12A2B2C2D2E3A3B3C3D3E4A4B4C4D6A6B6C6D6E6F6G6H6I6J6K12A12B12C12D
 size 116995424488181854542448812121212181818181818
ρ1111111111111111111111111111111    trivial
ρ211111111111-1-1-1-111111111111-1-1-1-1    linear of order 2
ρ311-111-111111-1-11111111-1-1-1-111-1-1-1-1    linear of order 2
ρ411-111-11111111-1-111111-1-1-1-1111111    linear of order 2
ρ511-1-1-1111111-ii-ii11111-1-1-1-1-1-1i-ii-i    linear of order 4
ρ611-1-1-1111111i-ii-i11111-1-1-1-1-1-1-ii-ii    linear of order 4
ρ7111-1-1-111111i-i-ii111111111-1-1-ii-ii    linear of order 4
ρ8111-1-1-111111-iii-i111111111-1-1i-ii-i    linear of order 4
ρ92-20-220222220000-2-2-2-2-200002-20000    orthogonal lifted from D4
ρ10220220-122-1-1-2-200-122-1-10000-1-11111    orthogonal lifted from D6
ρ11220220-122-1-12200-122-1-10000-1-1-1-1-1-1    orthogonal lifted from S3
ρ122-202-20222220000-2-2-2-2-20000-220000    orthogonal lifted from D4
ρ132-20-220-122-1-100001-2-2110000-11√3√3-√3-√3    orthogonal lifted from D12
ρ142-20-220-122-1-100001-2-2110000-11-√3-√3√3√3    orthogonal lifted from D12
ρ15220-2-20-122-1-12i-2i00-122-1-1000011i-ii-i    complex lifted from C4×S3
ρ16220-2-20-122-1-1-2i2i00-122-1-1000011-ii-ii    complex lifted from C4×S3
ρ172-202-20-122-1-100001-2-21100001-1√-3-√-3-√-3√-3    complex lifted from C3⋊D4
ρ182-202-20-122-1-100001-2-21100001-1-√-3√-3√-3-√-3    complex lifted from C3⋊D4
ρ1944-400041-2-2100004-21-21-1-122000000    orthogonal lifted from C2×C32⋊C4
ρ204440004-211-2000041-21-2-2-211000000    orthogonal lifted from C32⋊C4
ρ2144400041-2-2100004-21-2111-2-2000000    orthogonal lifted from C32⋊C4
ρ224-400004-211-20000-4-12-12003-3000000    orthogonal lifted from C62⋊C4
ρ234-400004-211-20000-4-12-1200-33000000    orthogonal lifted from C62⋊C4
ρ244-4000041-2-210000-42-12-1-3300000000    orthogonal lifted from C62⋊C4
ρ254-4000041-2-210000-42-12-13-300000000    orthogonal lifted from C62⋊C4
ρ2644-40004-211-2000041-21-222-1-1000000    orthogonal lifted from C2×C32⋊C4
ρ27880000-4-42-120000-42-4-120000000000    orthogonal lifted from S3×C32⋊C4
ρ288-80000-4-42-1200004-241-20000000000    orthogonal faithful
ρ29880000-42-42-10000-4-422-10000000000    orthogonal lifted from S3×C32⋊C4
ρ308-80000-42-42-1000044-2-210000000000    orthogonal faithful

Permutation representations of D6⋊(C32⋊C4)
►On 24 points - transitive group 24T1311
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 12)(2 11)(3 10)(4 9)(5 8)(6 7)(13 21)(14 20)(15 19)(16 24)(17 23)(18 22)

(13 17 15)(14 18 16)(19 21 23)(20 22 24)

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

(1 19 7 13)(2 20 8 14)(3 21 9 15)(4 22 10 16)(5 23 11 17)(6 24 12 18)
 
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), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,21)(14,20)(15,19)(16,24)(17,23)(18,22), (13,17,15)(14,18,16)(19,21,23)(20,22,24), (1,3,5)(2,4,6)(7,11,9)(8,12,10)(13,17,15)(14,18,16)(19,21,23)(20,22,24), (1,19,7,13)(2,20,8,14)(3,21,9,15)(4,22,10,16)(5,23,11,17)(6,24,12,18)>;
 
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), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,21)(14,20)(15,19)(16,24)(17,23)(18,22), (13,17,15)(14,18,16)(19,21,23)(20,22,24), (1,3,5)(2,4,6)(7,11,9)(8,12,10)(13,17,15)(14,18,16)(19,21,23)(20,22,24), (1,19,7,13)(2,20,8,14)(3,21,9,15)(4,22,10,16)(5,23,11,17)(6,24,12,18) );
 
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)], [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,21),(14,20),(15,19),(16,24),(17,23),(18,22)], [(13,17,15),(14,18,16),(19,21,23),(20,22,24)], [(1,3,5),(2,4,6),(7,11,9),(8,12,10),(13,17,15),(14,18,16),(19,21,23),(20,22,24)], [(1,19,7,13),(2,20,8,14),(3,21,9,15),(4,22,10,16),(5,23,11,17),(6,24,12,18)]])
 
G:=TransitiveGroup(24,1311);
 

Matrix representation of D6⋊(C32⋊C4) ►in GL6(𝔽13)

1120000
100000
001000
000100
000010
000001
,
1200000
1210000
001000
000100
000010
000001
,
100000
010000
0001012
001211212
0001200
0012100
,
100000
010000
0012100
0012000
00001212
000010
,
1060000
730000
0012000
000010
0011200
000011

G:=sub<GL(6,GF(13))| [1,1,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,12,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,12,0,0,1,1,12,1,0,0,0,12,0,0,0,0,12,12,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,1,0,0,0,0,0,0,0,12,1,0,0,0,0,12,0],[10,7,0,0,0,0,6,3,0,0,0,0,0,0,12,0,1,0,0,0,0,0,12,0,0,0,0,1,0,1,0,0,0,0,0,1] >;
 

D6⋊(C32⋊C4) in GAP, Magma, Sage, TeX 

D_6\rtimes (C_3^2\rtimes C_4)
 
% in TeX
 
G:=Group("D6:(C3^2:C4)");
 
// GroupNames label
 
G:=SmallGroup(432,568);
 
// by ID
 
G=gap.SmallGroup(432,568);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,141,36,1411,298,1356,1027,14118]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^6=b^2=c^3=d^3=e^4=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,e*b*e^-1=a^3*b,e*d*e^-1=c*d=d*c,e*c*e^-1=c^-1*d>;
 
// generators/relations
 

Export

Character table of D6⋊(C32⋊C4) in TeX

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