C3:S3:3C8 - GroupNames

G = C₃:S₃:₃C₈ order 144 = 2⁴·3²

2^nd semidirect product of C₃:S₃ and C₈ acting via C₈/C₄=C₂

Aliases: C₃:S₃:₃C₈, C₃²:₂(C₂xC₈), (C₃xC₁₂).₂C₄, C₃²:₂C₈:₅C₂, C₄.₃(C₃²:C₄), C₃:Dic₃.₇C₂², (C₄xC₃:S₃).₆C₂, (C₂xC₃:S₃).₅C₄, (C₃xC₆).₁(C₂xC₄), C₂.₁(C₂xC₃²:C₄), SmallGroup(144,130)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃² — C₃:S₃:₃C₈

Chief series

C₁ — C₃² — C₃xC₆ — C₃:Dic₃ — C₃²:₂C₈ — C₃:S₃:₃C₈

Lower central

C₃² — C₃:S₃:₃C₈

Upper central

C₁ — C₄

Generators and relations for C₃:S₃:₃C₈
G = < a,b,c,d | a³=b³=c²=d⁸=1, ab=ba, cac=a^-1, dad^-1=ab^-1, cbc=b^-1, dbd^-1=a^-1b^-1, cd=dc >

Subgroups: 158 in 38 conjugacy classes, 14 normal (10 characteristic)
Quotients: C₁, C₂, C₄, C₂², C₈, C₂xC₄, C₂xC₈, C₃²:C₄, C₂xC₃²:C₄, C₃:S₃:₃C₈

C₁

C₂

₉C₂

₂C₃

C₄

₉C₄

₉C₂²

₂C₆

₆S₃

C₃²

₉C₈

₉C₂xC₄

₂C₁₂

₆D₆

₆Dic₃

₆D₆

C₃:S₃

C₃xC₆

C₃:S₃

₉C₂xC₈

₆C₄xS₃

C₃:Dic₃

C₃xC₁₂

C₂xC₃:S₃

C₃²:₂C₈

C₄xC₃:S₃

C₃²:₂C₈

C₃:S₃:₃C₈

Character table of C₃:S₃:₃C₈

class	1	2A	2B	2C	3A	3B	4A	4B	4C	4D	6A	6B	8A	8B	8C	8D	8E	8F	8G	8H	12A	12B	12C	12D
size	1	1	9	9	4	4	1	1	9	9	4	4	9	9	9	9	9	9	9	9	4	4	4	4
ρ₁	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	1	1	1	1	1	1	1	1	1	1	-1	-1	-1	-1	-1	-1	-1	-1	1	1	1	1	linear of order 2
ρ₃	1	1	-1	-1	1	1	-1	-1	1	1	1	1	-1	1	1	1	-1	-1	-1	1	-1	-1	-1	-1	linear of order 2
ρ₄	1	1	-1	-1	1	1	-1	-1	1	1	1	1	1	-1	-1	-1	1	1	1	-1	-1	-1	-1	-1	linear of order 2
ρ₅	1	1	-1	-1	1	1	1	1	-1	-1	1	1	i	-i	i	i	-i	-i	i	-i	1	1	1	1	linear of order 4
ρ₆	1	1	1	1	1	1	-1	-1	-1	-1	1	1	i	i	-i	-i	-i	-i	i	i	-1	-1	-1	-1	linear of order 4
ρ₇	1	1	-1	-1	1	1	1	1	-1	-1	1	1	-i	i	-i	-i	i	i	-i	i	1	1	1	1	linear of order 4
ρ₈	1	1	1	1	1	1	-1	-1	-1	-1	1	1	-i	-i	i	i	i	i	-i	-i	-1	-1	-1	-1	linear of order 4
ρ₉	1	-1	1	-1	1	1	i	-i	i	-i	-1	-1	ζ₈	ζ₈⁵	ζ₈³	ζ₈⁷	ζ₈³	ζ₈⁷	ζ₈⁵	ζ₈	-i	-i	i	i	linear of order 8
ρ₁₀	1	-1	-1	1	1	1	i	-i	-i	i	-1	-1	ζ₈⁷	ζ₈⁷	ζ₈	ζ₈⁵	ζ₈⁵	ζ₈	ζ₈³	ζ₈³	-i	-i	i	i	linear of order 8
ρ₁₁	1	-1	-1	1	1	1	-i	i	i	-i	-1	-1	ζ₈	ζ₈	ζ₈⁷	ζ₈³	ζ₈³	ζ₈⁷	ζ₈⁵	ζ₈⁵	i	i	-i	-i	linear of order 8
ρ₁₂	1	-1	1	-1	1	1	-i	i	-i	i	-1	-1	ζ₈⁷	ζ₈³	ζ₈⁵	ζ₈	ζ₈⁵	ζ₈	ζ₈³	ζ₈⁷	i	i	-i	-i	linear of order 8
ρ₁₃	1	-1	-1	1	1	1	i	-i	-i	i	-1	-1	ζ₈³	ζ₈³	ζ₈⁵	ζ₈	ζ₈	ζ₈⁵	ζ₈⁷	ζ₈⁷	-i	-i	i	i	linear of order 8
ρ₁₄	1	-1	-1	1	1	1	-i	i	i	-i	-1	-1	ζ₈⁵	ζ₈⁵	ζ₈³	ζ₈⁷	ζ₈⁷	ζ₈³	ζ₈	ζ₈	i	i	-i	-i	linear of order 8
ρ₁₅	1	-1	1	-1	1	1	i	-i	i	-i	-1	-1	ζ₈⁵	ζ₈	ζ₈⁷	ζ₈³	ζ₈⁷	ζ₈³	ζ₈	ζ₈⁵	-i	-i	i	i	linear of order 8
ρ₁₆	1	-1	1	-1	1	1	-i	i	-i	i	-1	-1	ζ₈³	ζ₈⁷	ζ₈	ζ₈⁵	ζ₈	ζ₈⁵	ζ₈⁷	ζ₈³	i	i	-i	-i	linear of order 8
ρ₁₇	4	4	0	0	-2	1	4	4	0	0	-2	1	0	0	0	0	0	0	0	0	-2	1	1	-2	orthogonal lifted from C₃²:C₄
ρ₁₈	4	4	0	0	-2	1	-4	-4	0	0	-2	1	0	0	0	0	0	0	0	0	2	-1	-1	2	orthogonal lifted from C₂xC₃²:C₄
ρ₁₉	4	4	0	0	1	-2	4	4	0	0	1	-2	0	0	0	0	0	0	0	0	1	-2	-2	1	orthogonal lifted from C₃²:C₄
ρ₂₀	4	4	0	0	1	-2	-4	-4	0	0	1	-2	0	0	0	0	0	0	0	0	-1	2	2	-1	orthogonal lifted from C₂xC₃²:C₄
ρ₂₁	4	-4	0	0	1	-2	-4i	4i	0	0	-1	2	0	0	0	0	0	0	0	0	i	-2i	2i	-i	complex faithful
ρ₂₂	4	-4	0	0	1	-2	4i	-4i	0	0	-1	2	0	0	0	0	0	0	0	0	-i	2i	-2i	i	complex faithful
ρ₂₃	4	-4	0	0	-2	1	-4i	4i	0	0	2	-1	0	0	0	0	0	0	0	0	-2i	i	-i	2i	complex faithful
ρ₂₄	4	-4	0	0	-2	1	4i	-4i	0	0	2	-1	0	0	0	0	0	0	0	0	2i	-i	i	-2i	complex faithful

Permutation representations of C₃:S₃:₃C₈
►On 24 points - transitive group 24T244

Generators in S₂₄

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

(2 14 22)(4 24 16)(6 10 18)(8 20 12)

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

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

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

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

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

G:=TransitiveGroup(24,244);

C₃:S₃:₃C₈ is a maximal subgroup of
S₃²:C₈ C₄.₁₉S₃wrC₂ C₄.₄PSU₃(F₂) C₄.PSU₃(F₂) C₄.₂PSU₃(F₂) C₈xC₃²:C₄ (C₃xC₂₄):C₄ C₃:S₃.₅D₈ C₃:S₃.₅Q₁₆ C₄.₃F₉ C₄.F₉ C₃²:D₈:₅C₂ C₃:S₃:D₈ C₃:S₃:₂SD₁₆ C₃:S₃:Q₁₆ C₃:S₃:M₄(2) C₆².(C₂xC₄) C₁₂:S₃.C₄ C₃³:₅(C₂xC₈) C₃³:₇(C₂xC₈)
C₃:S₃:₃C₈ is a maximal quotient of
C₃:S₃:₃C₁₆ C₃²:₃M₅(2) C₄xC₃²:₂C₈ C₆².₆(C₂xC₄) C₃²:₅(C₄:C₈) He₃:₂(C₂xC₈) C₃³:₅(C₂xC₈) C₃³:₇(C₂xC₈)

Matrix representation of C₃:S₃:₃C₈ ►in GL₄(F₅) generated by

2	1	3	0
1	2	1	4
4	0	0	2
4	3	1	4

0	0	4	3
3	0	0	0
0	3	0	3
0	0	0	1

2	4	2	3
1	3	4	0
0	2	4	3
1	2	4	1

1	0	0	2
3	0	0	4
0	0	3	1
0	1	3	1

G:=sub<GL(4,GF(5))| [2,1,4,4,1,2,0,3,3,1,0,1,0,4,2,4],[0,3,0,0,0,0,3,0,4,0,0,0,3,0,3,1],[2,1,0,1,4,3,2,2,2,4,4,4,3,0,3,1],[1,3,0,0,0,0,0,1,0,0,3,3,2,4,1,1] >;

C₃:S₃:₃C₈ in GAP, Magma, Sage, TeX

C_3\rtimes S_3\rtimes_3C_8

% in TeX

G:=Group("C3:S3:3C8");

// GroupNames label

G:=SmallGroup(144,130);

// by ID

G=gap.SmallGroup(144,130);

# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,3,24,55,50,3364,256,4613,881]);

// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^2=d^8=1,a*b=b*a,c*a*c=a^-1,d*a*d^-1=a*b^-1,c*b*c=b^-1,d*b*d^-1=a^-1*b^-1,c*d=d*c>;

// generators/relations

Export

Subgroup lattice of C₃:S₃:₃C₈ in TeX
Character table of C₃:S₃:₃C₈ in TeX

G = C3:S3:3C8 order 144 = 24·32

2nd semidirect product of C3:S3 and C8 acting via C8/C4=C2

G = C₃:S₃:₃C₈ order 144 = 2⁴·3²

2^nd semidirect product of C₃:S₃ and C₈ acting via C₈/C₄=C₂