C2^3xS4 - GroupNames

G = C₂³×S₄ order 192 = 2⁶·3

Direct product of C₂³ and S₄

direct product, non-abelian, soluble, monomial, rational

Aliases: C₂³×S₄, A₄⋊C₂⁴, C₂⁵⋊₂S₃, C₂⁴⋊₄D₆, (C₂×A₄)⋊C₂³, C₂²⋊(S₃×C₂³), C₂³⋊(C₂²×S₃), (C₂³×A₄)⋊₃C₂, (C₂²×A₄)⋊₄C₂², SmallGroup(192,1537)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂² — A₄ — C₂³×S₄

Chief series

C₁ — C₂² — A₄ — S₄ — C₂×S₄ — C₂²×S₄ — C₂³×S₄

Lower central

A₄ — C₂³×S₄

Upper central

C₁ — C₂³

Generators and relations for C₂³×S₄
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=e²=f³=g²=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, bd=db, be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, fdf^-1=gdg=de=ed, fef^-1=d, eg=ge, gfg=f^-1 >

Subgroups: 2398 in 701 conjugacy classes, 99 normal (7 characteristic)
C₁, C₂, C₂, C₃, C₄, C₂², C₂², C₂², S₃, C₆, C₂×C₄, D₄, C₂³, C₂³, C₂³, A₄, D₆, C₂×C₆, C₂²×C₄, C₂×D₄, C₂⁴, C₂⁴, S₄, C₂×A₄, C₂²×S₃, C₂²×C₆, C₂³×C₄, C₂²×D₄, C₂⁵, C₂⁵, C₂×S₄, C₂²×A₄, S₃×C₂³, D₄×C₂³, C₂²×S₄, C₂³×A₄, C₂³×S₄
Quotients: C₁, C₂, C₂², S₃, C₂³, D₆, C₂⁴, S₄, C₂²×S₃, C₂×S₄, S₃×C₂³, C₂²×S₄, C₂³×S₄

Permutation representations of C₂³×S₄
►On 24 points - transitive group 24T400

Generators in S₂₄

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

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

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

(1 9)(2 7)(4 24)(6 23)(11 15)(12 13)(17 21)(18 19)

(2 7)(3 8)(4 24)(5 22)(10 14)(12 13)(16 20)(18 19)

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

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

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

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

G:=TransitiveGroup(24,400);

40 conjugacy classes

class	1	2A	···	2G	2H	···	2O	2P	···	2W	3	4A	···	4H	6A	···	6G
order	1	2	···	2	2	···	2	2	···	2	3	4	···	4	6	···	6
size	1	1	···	1	3	···	3	6	···	6	8	6	···	6	8	···	8

40 irreducible representations

dim	1	1	1	2	2	3	3
type	+	+	+	+	+	+	+
image	C₁	C₂	C₂	S₃	D₆	S₄	C₂×S₄
kernel	C₂³×S₄	C₂²×S₄	C₂³×A₄	C₂⁵	C₂⁴	C₂³	C₂²
# reps	1	14	1	1	7	2	14

Matrix representation of C₂³×S₄ ►in GL₇(ℤ)

-1	0	0	0	0	0	0
0	-1	0	0	0	0	0
0	0	1	0	0	0	0
0	0	0	1	0	0	0
0	0	0	0	-1	0	0
0	0	0	0	0	-1	0
0	0	0	0	0	0	-1

-1	0	0	0	0	0	0
0	-1	0	0	0	0	0
0	0	1	0	0	0	0
0	0	0	1	0	0	0
0	0	0	0	1	0	0
0	0	0	0	0	1	0
0	0	0	0	0	0	1

-1	0	0	0	0	0	0
0	-1	0	0	0	0	0
0	0	-1	0	0	0	0
0	0	0	-1	0	0	0
0	0	0	0	1	0	0
0	0	0	0	0	1	0
0	0	0	0	0	0	1

1	0	0	0	0	0	0
0	1	0	0	0	0	0
0	0	1	0	0	0	0
0	0	0	1	0	0	0
0	0	0	0	-1	0	0
0	0	0	0	0	-1	0
0	0	0	0	0	0	1

1	0	0	0	0	0	0
0	1	0	0	0	0	0
0	0	1	0	0	0	0
0	0	0	1	0	0	0
0	0	0	0	1	0	0
0	0	0	0	0	-1	0
0	0	0	0	0	0	-1

-1	-1	0	0	0	0	0
1	0	0	0	0	0	0
0	0	-1	-1	0	0	0
0	0	1	0	0	0	0
0	0	0	0	0	0	1
0	0	0	0	1	0	0
0	0	0	0	0	1	0

-1	0	0	0	0	0	0
1	1	0	0	0	0	0
0	0	-1	0	0	0	0
0	0	1	1	0	0	0
0	0	0	0	-1	0	0
0	0	0	0	0	0	-1
0	0	0	0	0	-1	0

G:=sub<GL(7,Integers())| [-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1],[-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1],[-1,1,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0],[-1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,-1,0] >;

C₂³×S₄ in GAP, Magma, Sage, TeX

C_2^3\times S_4

% in TeX

G:=Group("C2^3xS4");

// GroupNames label

G:=SmallGroup(192,1537);

// by ID

G=gap.SmallGroup(192,1537);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,2,1124,4037,285,2358,475]);

// Polycyclic

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

// generators/relations

G = C23×S4 order 192 = 26·3

Direct product of C23 and S4

G = C₂³×S₄ order 192 = 2⁶·3

Direct product of C₂³ and S₄