S3xM4(2) - GroupNames

G = S₃×M₄(2) order 96 = 2⁵·3

Direct product of S₃ and M₄(2)

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: S₃×M₄(2), C₈⋊₆D₆, C₂₄⋊₇C₂², Dic₃ ○M₄(2), C₁₂.₃₈C₂³, (S₃×C₈)⋊₇C₂, C₈⋊S₃⋊₅C₂, (C₄×S₃).₁C₄, C₄.₁₅(C₄×S₃), C₃⋊C₈⋊₁₁C₂², D₆.₆(C₂×C₄), (C₂×C₄).₄₅D₆, C₃⋊₂(C₂×M₄(2)), C₁₂.₁₂(C₂×C₄), C₄.Dic₃⋊₅C₂, C₂².₇(C₄×S₃), (C₃×M₄(2))⋊₅C₂, (C₂²×S₃).₄C₄, C₆.₁₅(C₂²×C₄), C₄.₃₈(C₂²×S₃), (C₂×Dic₃).₆C₄, Dic₃.₇(C₂×C₄), (C₄×S₃).₁₈C₂², (C₂×C₁₂).₂₅C₂², (S₃×C₂×C₄).₄C₂, C₂.₁₆(S₃×C₂×C₄), (C₂×C₆).₅(C₂×C₄), SmallGroup(96,113)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₆ — S₃×M₄(2)

Chief series

C₁ — C₃ — C₆ — C₁₂ — C₄×S₃ — S₃×C₂×C₄ — S₃×M₄(2)

Lower central

C₃ — C₆ — S₃×M₄(2)

Upper central

C₁ — C₄ — M₄(2)

Generators and relations for S₃×M₄(2)
G = < a,b,c,d | a³=b²=c⁸=d²=1, bab=a^-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c⁵ >

Subgroups: 130 in 68 conjugacy classes, 39 normal (27 characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₂², C₂², S₃, S₃, C₆, C₆, C₈, C₈, C₂×C₄, C₂×C₄, C₂³, Dic₃, C₁₂, D₆, D₆, C₂×C₆, C₂×C₈, M₄(2), M₄(2), C₂²×C₄, C₃⋊C₈, C₂₄, C₄×S₃, C₂×Dic₃, C₂×C₁₂, C₂²×S₃, C₂×M₄(2), S₃×C₈, C₈⋊S₃, C₄.Dic₃, C₃×M₄(2), S₃×C₂×C₄, S₃×M₄(2)
Quotients: C₁, C₂, C₄, C₂², S₃, C₂×C₄, C₂³, D₆, M₄(2), C₂²×C₄, C₄×S₃, C₂²×S₃, C₂×M₄(2), S₃×C₂×C₄, S₃×M₄(2)

Character table of S₃×M₄(2)

class	1	2A	2B	2C	2D	2E	3	4A	4B	4C	4D	4E	4F	6A	6B	8A	8B	8C	8D	8E	8F	8G	8H	12A	12B	12C	24A	24B	24C	24D
size	1	1	2	3	3	6	2	1	1	2	3	3	6	2	4	2	2	2	2	6	6	6	6	2	2	4	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	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	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
ρ₆	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
ρ₇	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
ρ₈	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
ρ₉	1	1	-1	1	1	-1	1	-1	-1	1	-1	-1	1	1	-1	i	-i	i	-i	-i	i	i	-i	-1	-1	1	-i	-i	i	i	linear of order 4
ρ₁₀	1	1	1	1	1	1	1	-1	-1	-1	-1	-1	-1	1	1	i	-i	-i	i	i	i	-i	-i	-1	-1	-1	i	-i	-i	i	linear of order 4
ρ₁₁	1	1	1	-1	-1	-1	1	-1	-1	-1	1	1	1	1	1	-i	i	i	-i	i	i	-i	-i	-1	-1	-1	-i	i	i	-i	linear of order 4
ρ₁₂	1	1	-1	-1	-1	1	1	-1	-1	1	1	1	-1	1	-1	-i	i	-i	i	-i	i	i	-i	-1	-1	1	i	i	-i	-i	linear of order 4
ρ₁₃	1	1	1	1	1	1	1	-1	-1	-1	-1	-1	-1	1	1	-i	i	i	-i	-i	-i	i	i	-1	-1	-1	-i	i	i	-i	linear of order 4
ρ₁₄	1	1	-1	1	1	-1	1	-1	-1	1	-1	-1	1	1	-1	-i	i	-i	i	i	-i	-i	i	-1	-1	1	i	i	-i	-i	linear of order 4
ρ₁₅	1	1	-1	-1	-1	1	1	-1	-1	1	1	1	-1	1	-1	i	-i	i	-i	i	-i	-i	i	-1	-1	1	-i	-i	i	i	linear of order 4
ρ₁₆	1	1	1	-1	-1	-1	1	-1	-1	-1	1	1	1	1	1	i	-i	-i	i	-i	-i	i	i	-1	-1	-1	i	-i	-i	i	linear of order 4
ρ₁₇	2	2	-2	0	0	0	-1	2	2	-2	0	0	0	-1	1	-2	-2	2	2	0	0	0	0	-1	-1	1	-1	1	-1	1	orthogonal lifted from D₆
ρ₁₈	2	2	2	0	0	0	-1	2	2	2	0	0	0	-1	-1	-2	-2	-2	-2	0	0	0	0	-1	-1	-1	1	1	1	1	orthogonal lifted from D₆
ρ₁₉	2	2	-2	0	0	0	-1	2	2	-2	0	0	0	-1	1	2	2	-2	-2	0	0	0	0	-1	-1	1	1	-1	1	-1	orthogonal lifted from D₆
ρ₂₀	2	2	2	0	0	0	-1	2	2	2	0	0	0	-1	-1	2	2	2	2	0	0	0	0	-1	-1	-1	-1	-1	-1	-1	orthogonal lifted from S₃
ρ₂₁	2	2	-2	0	0	0	-1	-2	-2	2	0	0	0	-1	1	2i	-2i	2i	-2i	0	0	0	0	1	1	-1	i	i	-i	-i	complex lifted from C₄×S₃
ρ₂₂	2	-2	0	2	-2	0	2	-2i	2i	0	2i	-2i	0	-2	0	0	0	0	0	0	0	0	0	-2i	2i	0	0	0	0	0	complex lifted from M₄(2)
ρ₂₃	2	-2	0	-2	2	0	2	2i	-2i	0	2i	-2i	0	-2	0	0	0	0	0	0	0	0	0	2i	-2i	0	0	0	0	0	complex lifted from M₄(2)
ρ₂₄	2	2	2	0	0	0	-1	-2	-2	-2	0	0	0	-1	-1	2i	-2i	-2i	2i	0	0	0	0	1	1	1	-i	i	i	-i	complex lifted from C₄×S₃
ρ₂₅	2	2	-2	0	0	0	-1	-2	-2	2	0	0	0	-1	1	-2i	2i	-2i	2i	0	0	0	0	1	1	-1	-i	-i	i	i	complex lifted from C₄×S₃
ρ₂₆	2	-2	0	-2	2	0	2	-2i	2i	0	-2i	2i	0	-2	0	0	0	0	0	0	0	0	0	-2i	2i	0	0	0	0	0	complex lifted from M₄(2)
ρ₂₇	2	2	2	0	0	0	-1	-2	-2	-2	0	0	0	-1	-1	-2i	2i	2i	-2i	0	0	0	0	1	1	1	i	-i	-i	i	complex lifted from C₄×S₃
ρ₂₈	2	-2	0	2	-2	0	2	2i	-2i	0	-2i	2i	0	-2	0	0	0	0	0	0	0	0	0	2i	-2i	0	0	0	0	0	complex lifted from M₄(2)
ρ₂₉	4	-4	0	0	0	0	-2	4i	-4i	0	0	0	0	2	0	0	0	0	0	0	0	0	0	-2i	2i	0	0	0	0	0	complex faithful
ρ₃₀	4	-4	0	0	0	0	-2	-4i	4i	0	0	0	0	2	0	0	0	0	0	0	0	0	0	2i	-2i	0	0	0	0	0	complex faithful

Permutation representations of S₃×M₄(2)
►On 24 points - transitive group 24T104

Generators in S₂₄

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

(9 22)(10 23)(11 24)(12 17)(13 18)(14 19)(15 20)(16 21)

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

(2 6)(4 8)(10 14)(12 16)(17 21)(19 23)

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

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

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

G:=TransitiveGroup(24,104);

S₃×M₄(2) is a maximal subgroup of
M₄(2).₁₉D₆ M₄(2).₂₁D₆ C₄²⋊₃D₆ M₄(2).₂₅D₆ M₄(2)⋊₂₆D₆ M₄(2)⋊₂₈D₆ D₈⋊₄D₆ D₂₄⋊C₂² C₂₄⋊D₆ C₃⋊C₈⋊₂₀D₆ C₄₀⋊D₆ D₁₅⋊₄M₄(2) D₁₅⋊M₄(2) D₁₅⋊₂M₄(2)
S₃×M₄(2) is a maximal quotient of
C₂₄⋊Q₈ C₄².₁₈₂D₆ C₈⋊₉D₁₂ Dic₃⋊₅M₄(2) Dic₃.₅M₄(2) Dic₃.M₄(2) D₆⋊M₄(2) D₆⋊₂M₄(2) Dic₃⋊M₄(2) C₄².₂₇D₆ C₄².₂₀₀D₆ C₄².₂₀₂D₆ D₆⋊₃M₄(2) C₁₂⋊M₄(2) Dic₃⋊₄M₄(2) D₆⋊₆M₄(2) C₂₄⋊D₄ C₂₄⋊₂₁D₄ C₂₄⋊D₆ C₃⋊C₈⋊₂₀D₆ C₄₀⋊D₆ D₁₅⋊₄M₄(2) D₁₅⋊M₄(2) D₁₅⋊₂M₄(2)

Matrix representation of S₃×M₄(2) ►in GL₄(𝔽₅) generated by

0	0	0	2
0	4	2	0
0	2	0	0
2	0	0	4

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

0	3	0	0
1	0	0	0
3	0	0	1
0	1	3	0

1	0	0	0
0	4	0	0
0	0	4	0
0	0	0	1

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

S₃×M₄(2) in GAP, Magma, Sage, TeX

S_3\times M_4(2)

% in TeX

G:=Group("S3xM4(2)");

// GroupNames label

G:=SmallGroup(96,113);

// by ID

G=gap.SmallGroup(96,113);

# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,188,50,69,2309]);

// Polycyclic

G:=Group<a,b,c,d|a^3=b^2=c^8=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^5>;

// generators/relations

Export

Character table of S₃×M₄(2) in TeX

G = S3×M4(2) order 96 = 25·3

Direct product of S3 and M4(2)

G = S₃×M₄(2) order 96 = 2⁵·3

Direct product of S₃ and M₄(2)