C4^2:4S3 - GroupNames

G = C₄²:₄S₃ order 96 = 2⁵·3

3^rd semidirect product of C₄² and S₃ acting via S₃/C₃=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D₁₂:₁C₄, C₄²:₄S₃, Dic₆:₁C₄, C₄.₁₇D₁₂, C₁₂.₃₃D₄, C₃:₁C₄wrC₂, (C₄xC₁₂):₆C₂, C₄.₆(C₄xS₃), (C₂xC₆).₂₆D₄, (C₂xC₄).₆₆D₆, C₁₂.₁₆(C₂xC₄), C₂.₃(D₆:C₄), C₄oD₁₂.₁C₂, C₄.Dic₃:₁C₂, C₆.₁(C₂²:C₄), (C₂xC₁₂).₉₆C₂², C₂².₇(C₃:D₄), SmallGroup(96,12)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₂ — C₄²:₄S₃

Chief series

C₁ — C₃ — C₆ — C₂xC₆ — C₂xC₁₂ — C₄oD₁₂ — C₄²:₄S₃

Lower central

C₃ — C₆ — C₁₂ — C₄²:₄S₃

Upper central

C₁ — C₄ — C₂xC₄ — C₄²

Generators and relations for C₄²:₄S₃
G = < a,b,c,d | a⁴=b⁴=c³=d²=1, dad=ab=ba, ac=ca, bc=cb, dbd=b^-1, dcd=c^-1 >

Subgroups: 106 in 44 conjugacy classes, 19 normal (all characteristic)
Quotients: C₁, C₂, C₄, C₂², S₃, C₂xC₄, D₄, D₆, C₂²:C₄, C₄xS₃, D₁₂, C₃:D₄, C₄wrC₂, D₆:C₄, C₄²:₄S₃

C₁

C₂

₂C₂

₁₂C₂

C₃

C₄

C₂²

₂C₄

₆C₂²

₆C₄

C₆

₂C₆

₄S₃

C₂xC₄

₂C₂xC₄

₃D₄

₃Q₈

₆C₈

₆D₄

₆C₂xC₄

C₁₂

C₂xC₆

C₁₂

₂C₁₂

₂D₆

₂Dic₃

C₄²

₃M₄(2)

₃C₄oD₄

C₂xC₁₂

D₁₂

Dic₆

₂C₃:D₄

₂C₄xS₃

₂C₃:C₈

₂C₂xC₁₂

₃C₄wrC₂

C₄oD₁₂

C₄.Dic₃

C₄xC₁₂

C₄²:₄S₃

Character table of C₄²:₄S₃

class	1	2A	2B	2C	3	4A	4B	4C	4D	4E	4F	4G	4H	6A	6B	6C	8A	8B	12A	12B	12C	12D	12E	12F	12G	12H	12I	12J	12K	12L
size	1	1	2	12	2	1	1	2	2	2	2	2	12	2	2	2	12	12	2	2	2	2	2	2	2	2	2	2	2	2
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	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	-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	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	-1	-1	1	-1	-1	1	linear of order 2
ρ₅	1	1	-1	1	1	-1	-1	-i	1	i	i	-i	-1	1	-1	-1	-i	i	i	i	1	-i	-i	-1	-i	-i	1	i	i	-1	linear of order 4
ρ₆	1	1	-1	1	1	-1	-1	i	1	-i	-i	i	-1	1	-1	-1	i	-i	-i	-i	1	i	i	-1	i	i	1	-i	-i	-1	linear of order 4
ρ₇	1	1	-1	-1	1	-1	-1	-i	1	i	i	-i	1	1	-1	-1	i	-i	i	i	1	-i	-i	-1	-i	-i	1	i	i	-1	linear of order 4
ρ₈	1	1	-1	-1	1	-1	-1	i	1	-i	-i	i	1	1	-1	-1	-i	i	-i	-i	1	i	i	-1	i	i	1	-i	-i	-1	linear of order 4
ρ₉	2	2	2	0	2	-2	-2	0	-2	0	0	0	0	2	2	2	0	0	0	0	-2	0	0	-2	0	0	-2	0	0	-2	orthogonal lifted from D₄
ρ₁₀	2	2	2	0	-1	2	2	-2	2	-2	-2	-2	0	-1	-1	-1	0	0	1	1	-1	1	1	-1	1	1	-1	1	1	-1	orthogonal lifted from D₆
ρ₁₁	2	2	2	0	-1	2	2	2	2	2	2	2	0	-1	-1	-1	0	0	-1	-1	-1	-1	-1	-1	-1	-1	-1	-1	-1	-1	orthogonal lifted from S₃
ρ₁₂	2	2	-2	0	2	2	2	0	-2	0	0	0	0	2	-2	-2	0	0	0	0	-2	0	0	2	0	0	-2	0	0	2	orthogonal lifted from D₄
ρ₁₃	2	2	-2	0	-1	2	2	0	-2	0	0	0	0	-1	1	1	0	0	-√3	√3	1	√3	-√3	-1	√3	-√3	1	-√3	√3	-1	orthogonal lifted from D₁₂
ρ₁₄	2	2	-2	0	-1	2	2	0	-2	0	0	0	0	-1	1	1	0	0	√3	-√3	1	-√3	√3	-1	-√3	√3	1	√3	-√3	-1	orthogonal lifted from D₁₂
ρ₁₅	2	2	2	0	-1	-2	-2	0	-2	0	0	0	0	-1	-1	-1	0	0	-√-3	√-3	1	-√-3	√-3	1	-√-3	√-3	1	-√-3	√-3	1	complex lifted from C₃:D₄
ρ₁₆	2	-2	0	0	2	2i	-2i	1-i	0	-1-i	1+i	-1+i	0	-2	0	0	0	0	1+i	1+i	0	-1+i	-1+i	-2i	1-i	1-i	0	-1-i	-1-i	2i	complex lifted from C₄wrC₂
ρ₁₇	2	2	2	0	-1	-2	-2	0	-2	0	0	0	0	-1	-1	-1	0	0	√-3	-√-3	1	√-3	-√-3	1	√-3	-√-3	1	√-3	-√-3	1	complex lifted from C₃:D₄
ρ₁₈	2	2	-2	0	-1	-2	-2	2i	2	-2i	-2i	2i	0	-1	1	1	0	0	i	i	-1	-i	-i	1	-i	-i	-1	i	i	1	complex lifted from C₄xS₃
ρ₁₉	2	-2	0	0	2	-2i	2i	1+i	0	-1+i	1-i	-1-i	0	-2	0	0	0	0	1-i	1-i	0	-1-i	-1-i	2i	1+i	1+i	0	-1+i	-1+i	-2i	complex lifted from C₄wrC₂
ρ₂₀	2	2	-2	0	-1	-2	-2	-2i	2	2i	2i	-2i	0	-1	1	1	0	0	-i	-i	-1	i	i	1	i	i	-1	-i	-i	1	complex lifted from C₄xS₃
ρ₂₁	2	-2	0	0	2	-2i	2i	-1-i	0	1-i	-1+i	1+i	0	-2	0	0	0	0	-1+i	-1+i	0	1+i	1+i	2i	-1-i	-1-i	0	1-i	1-i	-2i	complex lifted from C₄wrC₂
ρ₂₂	2	-2	0	0	-1	2i	-2i	-1+i	0	1+i	-1-i	1-i	0	1	-√-3	√-3	0	0	ζ₄³ζ₃+ζ₃+1	ζ₄³ζ₃²+ζ₃²+1	√3	ζ₄ζ₃²+ζ₄+ζ₃²	ζ₄ζ₃+ζ₄+ζ₃	i	ζ₄ζ₃+ζ₃+1	ζ₄ζ₃²+ζ₃²+1	-√3	ζ₄³ζ₃²+ζ₄³+ζ₃²	ζ₄³ζ₃+ζ₄³+ζ₃	-i	complex faithful
ρ₂₃	2	-2	0	0	2	2i	-2i	-1+i	0	1+i	-1-i	1-i	0	-2	0	0	0	0	-1-i	-1-i	0	1-i	1-i	-2i	-1+i	-1+i	0	1+i	1+i	2i	complex lifted from C₄wrC₂
ρ₂₄	2	-2	0	0	-1	-2i	2i	1+i	0	-1+i	1-i	-1-i	0	1	√-3	-√-3	0	0	ζ₄ζ₃+ζ₄+ζ₃	ζ₄ζ₃²+ζ₄+ζ₃²	√3	ζ₄³ζ₃²+ζ₃²+1	ζ₄³ζ₃+ζ₃+1	-i	ζ₄³ζ₃+ζ₄³+ζ₃	ζ₄³ζ₃²+ζ₄³+ζ₃²	-√3	ζ₄ζ₃²+ζ₃²+1	ζ₄ζ₃+ζ₃+1	i	complex faithful
ρ₂₅	2	-2	0	0	-1	2i	-2i	1-i	0	-1-i	1+i	-1+i	0	1	√-3	-√-3	0	0	ζ₄³ζ₃+ζ₄³+ζ₃	ζ₄³ζ₃²+ζ₄³+ζ₃²	-√3	ζ₄ζ₃²+ζ₃²+1	ζ₄ζ₃+ζ₃+1	i	ζ₄ζ₃+ζ₄+ζ₃	ζ₄ζ₃²+ζ₄+ζ₃²	√3	ζ₄³ζ₃²+ζ₃²+1	ζ₄³ζ₃+ζ₃+1	-i	complex faithful
ρ₂₆	2	-2	0	0	-1	-2i	2i	-1-i	0	1-i	-1+i	1+i	0	1	√-3	-√-3	0	0	ζ₄ζ₃²+ζ₃²+1	ζ₄ζ₃+ζ₃+1	√3	ζ₄³ζ₃+ζ₄³+ζ₃	ζ₄³ζ₃²+ζ₄³+ζ₃²	-i	ζ₄³ζ₃²+ζ₃²+1	ζ₄³ζ₃+ζ₃+1	-√3	ζ₄ζ₃+ζ₄+ζ₃	ζ₄ζ₃²+ζ₄+ζ₃²	i	complex faithful
ρ₂₇	2	-2	0	0	-1	-2i	2i	1+i	0	-1+i	1-i	-1-i	0	1	-√-3	√-3	0	0	ζ₄ζ₃²+ζ₄+ζ₃²	ζ₄ζ₃+ζ₄+ζ₃	-√3	ζ₄³ζ₃+ζ₃+1	ζ₄³ζ₃²+ζ₃²+1	-i	ζ₄³ζ₃²+ζ₄³+ζ₃²	ζ₄³ζ₃+ζ₄³+ζ₃	√3	ζ₄ζ₃+ζ₃+1	ζ₄ζ₃²+ζ₃²+1	i	complex faithful
ρ₂₈	2	-2	0	0	-1	2i	-2i	1-i	0	-1-i	1+i	-1+i	0	1	-√-3	√-3	0	0	ζ₄³ζ₃²+ζ₄³+ζ₃²	ζ₄³ζ₃+ζ₄³+ζ₃	√3	ζ₄ζ₃+ζ₃+1	ζ₄ζ₃²+ζ₃²+1	i	ζ₄ζ₃²+ζ₄+ζ₃²	ζ₄ζ₃+ζ₄+ζ₃	-√3	ζ₄³ζ₃+ζ₃+1	ζ₄³ζ₃²+ζ₃²+1	-i	complex faithful
ρ₂₉	2	-2	0	0	-1	-2i	2i	-1-i	0	1-i	-1+i	1+i	0	1	-√-3	√-3	0	0	ζ₄ζ₃+ζ₃+1	ζ₄ζ₃²+ζ₃²+1	-√3	ζ₄³ζ₃²+ζ₄³+ζ₃²	ζ₄³ζ₃+ζ₄³+ζ₃	-i	ζ₄³ζ₃+ζ₃+1	ζ₄³ζ₃²+ζ₃²+1	√3	ζ₄ζ₃²+ζ₄+ζ₃²	ζ₄ζ₃+ζ₄+ζ₃	i	complex faithful
ρ₃₀	2	-2	0	0	-1	2i	-2i	-1+i	0	1+i	-1-i	1-i	0	1	√-3	-√-3	0	0	ζ₄³ζ₃²+ζ₃²+1	ζ₄³ζ₃+ζ₃+1	-√3	ζ₄ζ₃+ζ₄+ζ₃	ζ₄ζ₃²+ζ₄+ζ₃²	i	ζ₄ζ₃²+ζ₃²+1	ζ₄ζ₃+ζ₃+1	√3	ζ₄³ζ₃+ζ₄³+ζ₃	ζ₄³ζ₃²+ζ₄³+ζ₃²	-i	complex faithful

Permutation representations of C₄²:₄S₃
►On 24 points - transitive group 24T117

Generators in S₂₄

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

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

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

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

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

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

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

G:=TransitiveGroup(24,117);

C₄²:₄S₃ is a maximal subgroup of
D₂₄:₁₁C₄ D₂₄:₄C₄ S₃xC₄wrC₂ C₄²:₃D₆ Q₈:₅D₁₂ M₄(2).₂₂D₆ C₄².₁₉₆D₆ C₄²:₅D₆ Q₈.₁₄D₁₂ D₄.₁₀D₁₂ C₄²:₆D₆ C₄²:₇D₆ D₁₂.₁₄D₄ C₄²:₈D₆ D₁₂.₁₅D₄ C₄²:₄D₉ C₄²:D₉ D₁₂:₂Dic₃ C₁₂.₈₀D₁₂ C₁₂²:C₂ (C₄xC₁₂):S₃ C₆₀.₉₉D₄ D₆₀:₁₆C₄ D₆₀:₇C₄ D₁₂:₂F₅ D₆₀:₅C₄
C₄²:₄S₃ is a maximal quotient of
C₆.C₄wrC₂ C₄:Dic₃:C₄ C₄.₈Dic₁₂ C₄.₁₇D₂₄ C₄².D₆ C₄².₂D₆ C₁₂.₈C₄² C₄²:₄D₉ D₁₂:₂Dic₃ C₁₂.₈₀D₁₂ C₁₂²:C₂ C₆₀.₉₉D₄ D₆₀:₁₆C₄ D₆₀:₇C₄ D₁₂:₂F₅ D₆₀:₅C₄

Matrix representation of C₄²:₄S₃ ►in GL₂(F₁₃) generated by

8	0
0	1

5	0
0	8

9	0
0	3

0	1
1	0

G:=sub<GL(2,GF(13))| [8,0,0,1],[5,0,0,8],[9,0,0,3],[0,1,1,0] >;

C₄²:₄S₃ in GAP, Magma, Sage, TeX

C_4^2\rtimes_4S_3

% in TeX

G:=Group("C4^2:4S3");

// GroupNames label

G:=SmallGroup(96,12);

// by ID

G=gap.SmallGroup(96,12);

# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,121,31,362,579,69,2309]);

// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=c^3=d^2=1,d*a*d=a*b=b*a,a*c=c*a,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;

// generators/relations

Export

Subgroup lattice of C₄²:₄S₃ in TeX
Character table of C₄²:₄S₃ in TeX

G = C42:4S3 order 96 = 25·3

3rd semidirect product of C42 and S3 acting via S3/C3=C2

G = C₄²:₄S₃ order 96 = 2⁵·3

3^rd semidirect product of C₄² and S₃ acting via S₃/C₃=C₂