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G = C62⋊Dic3  order 432 = 24·33

The semidirect product of C62 and Dic3 acting faithfully

non-abelian, soluble, monomial

Aliases: C62⋊Dic3, C3⋊S3.2S4, A4⋊(C32⋊C4), (C32×A4)⋊1C4, C32⋊3(A4⋊C4), C22⋊(C33⋊C4), (A4×C3⋊S3).1C2, (C22×C3⋊S3).3S3, SmallGroup(432,743)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22 — C32×A4 — C62⋊Dic3
Chief series
C1 — C22 — C62 — C32×A4 — A4×C3⋊S3 — C62⋊Dic3
Lower central
C32×A4 — C62⋊Dic3
Upper central
C1

Generators and relations for C62⋊Dic3
 G = < a,b,c,d | a6=b6=c6=1, d2=c3, ab=ba, cac-1=a-1b3, dad-1=a4b-1, cbc-1=a3b2, dbd-1=a-1b2, dcd-1=c-1 >

Subgroups: 752 in 70 conjugacy classes, 10 normal (all characteristic)
C1, C2, C3, C4, C22, C22, S3, C6, C2×C4, C23, C32, C32, Dic3, A4, A4, D6, C2×C6, C22⋊C4, C3×S3, C3⋊S3, C3⋊S3, C3×C6, C2×A4, C22×S3, C33, C32⋊C4, C3×A4, C2×C3⋊S3, C62, A4⋊C4, C3×C3⋊S3, S3×A4, C2×C32⋊C4, C22×C3⋊S3, C33⋊C4, C32×A4, C62⋊C4, A4×C3⋊S3, C62⋊Dic3
Quotients: C1, C2, C4, S3, Dic3, S4, C32⋊C4, A4⋊C4, C33⋊C4, C62⋊Dic3

Character table of C62⋊Dic3

 class 12A2B2C3A3B3C3D3E3F3G4A4B4C4D6A6B6C
 size 139274481616161654545454121272
ρ1111111111111111111    trivial
ρ211111111111-1-1-1-1111    linear of order 2
ρ311-1-11111111-iii-i11-1    linear of order 4
ρ411-1-11111111i-i-ii11-1    linear of order 4
ρ5222222-1-1-1-1-1000022-1    orthogonal lifted from S3
ρ622-2-222-1-1-1-1-10000221    symplectic lifted from Dic3, Schur index 2
ρ73-13-13300000-11-11-1-10    orthogonal lifted from S4
ρ83-13-133000001-11-1-1-10    orthogonal lifted from S4
ρ93-1-313300000ii-i-i-1-10    complex lifted from A4⋊C4
ρ103-1-313300000-i-iii-1-10    complex lifted from A4⋊C4
ρ1144001-24-21-2100001-20    orthogonal lifted from C32⋊C4
ρ124400-2141-21-20000-210    orthogonal lifted from C32⋊C4
ρ1344001-2-21-1+3√-3/21-1-3√-3/200001-20    complex lifted from C33⋊C4
ρ144400-21-2-1-3√-3/21-1+3√-3/210000-210    complex lifted from C33⋊C4
ρ1544001-2-21-1-3√-3/21-1+3√-3/200001-20    complex lifted from C33⋊C4
ρ164400-21-2-1+3√-3/21-1-3√-3/210000-210    complex lifted from C33⋊C4
ρ1712-4003-6000000000-120    orthogonal faithful
ρ1812-400-630000000002-10    orthogonal faithful

Permutation representations of C62⋊Dic3
►On 24 points - transitive group 24T1339
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 17)(2 18)(3 13)(4 14)(5 15)(6 16)(7 22 11 20 9 24)(8 23 12 21 10 19)

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

(1 19)(2 10 6 8)(3 21 5 23)(4 12)(7 16 11 18)(9 14)(13 24 15 20)(17 22)
 
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), (1,17)(2,18)(3,13)(4,14)(5,15)(6,16)(7,22,11,20,9,24)(8,23,12,21,10,19), (2,13,18,6,15,16)(3,5)(4,17,14)(7,24,10,11,20,8)(9,22,12)(21,23), (1,19)(2,10,6,8)(3,21,5,23)(4,12)(7,16,11,18)(9,14)(13,24,15,20)(17,22)>;
 
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), (1,17)(2,18)(3,13)(4,14)(5,15)(6,16)(7,22,11,20,9,24)(8,23,12,21,10,19), (2,13,18,6,15,16)(3,5)(4,17,14)(7,24,10,11,20,8)(9,22,12)(21,23), (1,19)(2,10,6,8)(3,21,5,23)(4,12)(7,16,11,18)(9,14)(13,24,15,20)(17,22) );
 
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)], [(1,17),(2,18),(3,13),(4,14),(5,15),(6,16),(7,22,11,20,9,24),(8,23,12,21,10,19)], [(2,13,18,6,15,16),(3,5),(4,17,14),(7,24,10,11,20,8),(9,22,12),(21,23)], [(1,19),(2,10,6,8),(3,21,5,23),(4,12),(7,16,11,18),(9,14),(13,24,15,20),(17,22)]])
 
G:=TransitiveGroup(24,1339);
 

Matrix representation of C62⋊Dic3 ►in GL7(𝔽13)

01210000
01200000
11200000
00001200
00011200
0000414
0001212911
,
12000000
12010000
12100000
0001000
0000100
00099119
0001141
,
01200000
00120000
12000000
0000100
0001000
0000010
0001212912
,
0800000
8000000
0080000
0000001
0001212912
0000010
0000100

G:=sub<GL(7,GF(13))| [0,0,1,0,0,0,0,12,12,12,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,12,0,0,0,12,12,4,12,0,0,0,0,0,1,9,0,0,0,0,0,4,11],[12,12,12,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,9,1,0,0,0,0,1,9,1,0,0,0,0,0,11,4,0,0,0,0,0,9,1],[0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,12,0,0,0,1,0,0,12,0,0,0,0,0,1,9,0,0,0,0,0,0,12],[0,8,0,0,0,0,0,8,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,1,0,0,0,0,9,1,0,0,0,0,1,12,0,0] >;
 

C62⋊Dic3 in GAP, Magma, Sage, TeX 

C_6^2\rtimes {\rm Dic}_3
 
% in TeX
 
G:=Group("C6^2:Dic3");
 
// GroupNames label
 
G:=SmallGroup(432,743);
 
// by ID
 
G=gap.SmallGroup(432,743);
 
# by ID
 
G:=PCGroup([7,-2,-2,-3,-3,3,-2,2,14,170,1683,346,1684,1271,9077,2287,5298,3989]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^6=b^6=c^6=1,d^2=c^3,a*b=b*a,c*a*c^-1=a^-1*b^3,d*a*d^-1=a^4*b^-1,c*b*c^-1=a^3*b^2,d*b*d^-1=a^-1*b^2,d*c*d^-1=c^-1>;
 
// generators/relations
 

Export

Character table of C62⋊Dic3 in TeX

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