C2^2xS3^2 - GroupNames

G = C₂²×S₃² order 144 = 2⁴·3²

Direct product of C₂², S₃ and S₃

direct product, metabelian, supersoluble, monomial, A-group, rational

Aliases: C₂²×S₃², C₃²⋊C₂⁴, C₆²⋊₅C₂², C₃⋊S₃⋊C₂³, (C₂×C₆)⋊₈D₆, (C₃×C₆)⋊C₂³, (C₃×S₃)⋊C₂³, C₃⋊₁(S₃×C₂³), C₆⋊₁(C₂²×S₃), (S₃×C₆)⋊₁₀C₂², (S₃×C₂×C₆)⋊₇C₂, (C₂²×C₃⋊S₃)⋊₆C₂, (C₂×C₃⋊S₃)⋊₈C₂², SmallGroup(144,192)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃² — C₂²×S₃²

Chief series

C₁ — C₃ — C₃² — C₃×S₃ — S₃² — C₂×S₃² — C₂²×S₃²

Lower central

C₃² — C₂²×S₃²

Upper central

C₁ — C₂²

Generators and relations for C₂²×S₃²
G = < a,b,c,d,e,f | a²=b²=c³=d²=e³=f²=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd=c^-1, ce=ec, cf=fc, de=ed, df=fd, fef=e^-1 >

Subgroups: 912 in 284 conjugacy classes, 104 normal (6 characteristic)
C₁, C₂, C₂, C₃, C₃, C₂², C₂², S₃, S₃, C₆, C₆, C₂³, C₃², D₆, D₆, C₂×C₆, C₂×C₆, C₂⁴, C₃×S₃, C₃⋊S₃, C₃×C₆, C₂²×S₃, C₂²×S₃, C₂²×C₆, S₃², S₃×C₆, C₂×C₃⋊S₃, C₆², S₃×C₂³, C₂×S₃², S₃×C₂×C₆, C₂²×C₃⋊S₃, C₂²×S₃²
Quotients: C₁, C₂, C₂², S₃, C₂³, D₆, C₂⁴, C₂²×S₃, S₃², S₃×C₂³, C₂×S₃², C₂²×S₃²

Permutation representations of C₂²×S₃²
►On 24 points - transitive group 24T232

Generators in S₂₄

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

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

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

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

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

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

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

G:=TransitiveGroup(24,232);

C₂²×S₃² is a maximal subgroup of
C₆².₉₁C₂³ D₆⋊₄D₁₂ D₆⋊₅D₁₂ C₆².₁₂₅C₂³ D₆≀C₂
C₂²×S₃² is a maximal quotient of
D₁₂.₃₃D₆ D₁₂.₃₄D₆ D₁₂⋊₂₃D₆ D₁₂⋊₂₄D₆ D₁₂⋊₂₇D₆ Dic₆.₂₄D₆ Dic₆⋊₁₂D₆ D₁₂⋊₁₂D₆ D₁₂⋊₁₃D₆ D₁₂.₂₅D₆ Dic₆.₂₆D₆ D₁₂⋊₁₅D₆ D₁₂⋊₁₆D₆ C₃²⋊2₊¹⁺⁴

36 conjugacy classes

class	1	2A	2B	2C	2D	···	2K	2L	2M	2N	2O	3A	3B	3C	6A	···	6F	6G	6H	6I	6J	···	6Q
order	1	2	2	2	2	···	2	2	2	2	2	3	3	3	6	···	6	6	6	6	6	···	6
size	1	1	1	1	3	···	3	9	9	9	9	2	2	4	2	···	2	4	4	4	6	···	6

36 irreducible representations

dim	1	1	1	1	2	2	2	4	4
type	+	+	+	+	+	+	+	+	+
image	C₁	C₂	C₂	C₂	S₃	D₆	D₆	S₃²	C₂×S₃²
kernel	C₂²×S₃²	C₂×S₃²	S₃×C₂×C₆	C₂²×C₃⋊S₃	C₂²×S₃	D₆	C₂×C₆	C₂²	C₂
# reps	1	12	2	1	2	12	2	1	3

Matrix representation of C₂²×S₃² ►in GL₆(ℤ)

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

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

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

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	0	1
0	0	0	0	1	0

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

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

G:=sub<GL(6,Integers())| [-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,-1,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[-1,-1,0,0,0,0,1,0,0,0,0,0,0,0,-1,-1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C₂²×S₃² in GAP, Magma, Sage, TeX

C_2^2\times S_3^2

% in TeX

G:=Group("C2^2xS3^2");

// GroupNames label

G:=SmallGroup(144,192);

// by ID

G=gap.SmallGroup(144,192);

# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,490,3461]);

// Polycyclic

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

// generators/relations

G = C22×S32 order 144 = 24·32

Direct product of C22, S3 and S3

G = C₂²×S₃² order 144 = 2⁴·3²

Direct product of C₂², S₃ and S₃