C2^2xC3:S4

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

Aliases: C₂²×C₃⋊S₄, C₆⋊₂(C₂×S₄), (C₂×C₆)⋊₅S₄, (C₂×A₄)⋊₂D₆, (C₂³×C₆)⋊₅S₃, C₃⋊₂(C₂²×S₄), (C₂²×C₆)⋊₃D₆, C₂⁴⋊₃(C₃⋊S₃), (C₃×A₄)⋊₃C₂³, (C₂²×A₄)⋊₅S₃, (C₆×A₄)⋊₃C₂², A₄⋊₂(C₂²×S₃), (A₄×C₂×C₆)⋊₆C₂, C₂³⋊(C₂×C₃⋊S₃), C₂²⋊(C₂²×C₃⋊S₃), (C₂×C₆)⋊₃(C₂²×S₃), SmallGroup(288,1034)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂² — C₃×A₄ — C₂²×C₃⋊S₄

Chief series

C₁ — C₂² — C₂×C₆ — C₃×A₄ — C₃⋊S₄ — C₂×C₃⋊S₄ — C₂²×C₃⋊S₄

Lower central

C₃×A₄ — C₂²×C₃⋊S₄

Upper central

C₁ — C₂²

Subgroups: 1876 in 326 conjugacy classes, 51 normal (11 characteristic)
C₁, C₂^[×3], C₂^[×8], C₃, C₃^[×3], C₄^[×4], C₂²^[×2], C₂²^[×26], S₃^[×16], C₆^[×3], C₆^[×13], C₂×C₄^[×6], D₄^[×16], C₂³^[×3], C₂³^[×14], C₃², Dic₃^[×4], A₄^[×3], D₆^[×34], C₂×C₆^[×2], C₂×C₆^[×13], C₂²×C₄, C₂×D₄^[×12], C₂⁴, C₂⁴, C₃⋊S₃^[×4], C₃×C₆^[×3], C₂×Dic₃^[×6], C₃⋊D₄^[×16], S₄^[×12], C₂×A₄^[×9], C₂²×S₃^[×13], C₂²×C₆^[×3], C₂²×C₆^[×4], C₂²×D₄, C₃×A₄, C₂×C₃⋊S₃^[×6], C₆², C₂²×Dic₃, C₂×C₃⋊D₄^[×12], C₂×S₄^[×18], C₂²×A₄^[×3], S₃×C₂³, C₂³×C₆, C₃⋊S₄^[×4], C₆×A₄^[×3], C₂²×C₃⋊S₃, C₂²×C₃⋊D₄, C₂²×S₄^[×3], C₂×C₃⋊S₄^[×6], A₄×C₂×C₆, C₂²×C₃⋊S₄

Quotients:
C₁, C₂^[×7], C₂²^[×7], S₃^[×4], C₂³, D₆^[×12], C₃⋊S₃, S₄, C₂²×S₃^[×4], C₂×C₃⋊S₃^[×3], C₂×S₄^[×3], C₃⋊S₄, C₂²×C₃⋊S₃, C₂²×S₄, C₂×C₃⋊S₄^[×3], C₂²×C₃⋊S₄

Generators and relations
G = < a,b,c,d,e,f,g | a²=b²=c³=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, gcg=c^-1, fdf^-1=gdg=de=ed, fef^-1=d, eg=ge, gfg=f^-1 >

Smallest permutation representation
►On 36 points

Generators in S₃₆

(1 25)(2 26)(3 27)(4 16)(5 17)(6 18)(7 22)(8 23)(9 24)(10 31)(11 32)(12 33)(13 28)(14 29)(15 30)(19 34)(20 35)(21 36)

(1 22)(2 23)(3 24)(4 19)(5 20)(6 21)(7 25)(8 26)(9 27)(10 28)(11 29)(12 30)(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)

(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)(25 26 27)(28 29 30)(31 32 33)(34 35 36)

(1 7)(2 8)(3 9)(10 13)(11 14)(12 15)(22 25)(23 26)(24 27)(28 31)(29 32)(30 33)

(4 34)(5 35)(6 36)(10 13)(11 14)(12 15)(16 19)(17 20)(18 21)(28 31)(29 32)(30 33)

(1 10 16)(2 11 17)(3 12 18)(4 25 31)(5 26 32)(6 27 33)(7 13 19)(8 14 20)(9 15 21)(22 28 34)(23 29 35)(24 30 36)

(1 7)(2 9)(3 8)(4 28)(5 30)(6 29)(10 19)(11 21)(12 20)(13 16)(14 18)(15 17)(22 25)(23 27)(24 26)(31 34)(32 36)(33 35)

G:=sub<Sym(36)| (1,25)(2,26)(3,27)(4,16)(5,17)(6,18)(7,22)(8,23)(9,24)(10,31)(11,32)(12,33)(13,28)(14,29)(15,30)(19,34)(20,35)(21,36), (1,22)(2,23)(3,24)(4,19)(5,20)(6,21)(7,25)(8,26)(9,27)(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24)(25,26,27)(28,29,30)(31,32,33)(34,35,36), (1,7)(2,8)(3,9)(10,13)(11,14)(12,15)(22,25)(23,26)(24,27)(28,31)(29,32)(30,33), (4,34)(5,35)(6,36)(10,13)(11,14)(12,15)(16,19)(17,20)(18,21)(28,31)(29,32)(30,33), (1,10,16)(2,11,17)(3,12,18)(4,25,31)(5,26,32)(6,27,33)(7,13,19)(8,14,20)(9,15,21)(22,28,34)(23,29,35)(24,30,36), (1,7)(2,9)(3,8)(4,28)(5,30)(6,29)(10,19)(11,21)(12,20)(13,16)(14,18)(15,17)(22,25)(23,27)(24,26)(31,34)(32,36)(33,35)>;

G:=Group( (1,25)(2,26)(3,27)(4,16)(5,17)(6,18)(7,22)(8,23)(9,24)(10,31)(11,32)(12,33)(13,28)(14,29)(15,30)(19,34)(20,35)(21,36), (1,22)(2,23)(3,24)(4,19)(5,20)(6,21)(7,25)(8,26)(9,27)(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24)(25,26,27)(28,29,30)(31,32,33)(34,35,36), (1,7)(2,8)(3,9)(10,13)(11,14)(12,15)(22,25)(23,26)(24,27)(28,31)(29,32)(30,33), (4,34)(5,35)(6,36)(10,13)(11,14)(12,15)(16,19)(17,20)(18,21)(28,31)(29,32)(30,33), (1,10,16)(2,11,17)(3,12,18)(4,25,31)(5,26,32)(6,27,33)(7,13,19)(8,14,20)(9,15,21)(22,28,34)(23,29,35)(24,30,36), (1,7)(2,9)(3,8)(4,28)(5,30)(6,29)(10,19)(11,21)(12,20)(13,16)(14,18)(15,17)(22,25)(23,27)(24,26)(31,34)(32,36)(33,35) );

G=PermutationGroup([(1,25),(2,26),(3,27),(4,16),(5,17),(6,18),(7,22),(8,23),(9,24),(10,31),(11,32),(12,33),(13,28),(14,29),(15,30),(19,34),(20,35),(21,36)], [(1,22),(2,23),(3,24),(4,19),(5,20),(6,21),(7,25),(8,26),(9,27),(10,28),(11,29),(12,30),(13,31),(14,32),(15,33),(16,34),(17,35),(18,36)], [(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15),(16,17,18),(19,20,21),(22,23,24),(25,26,27),(28,29,30),(31,32,33),(34,35,36)], [(1,7),(2,8),(3,9),(10,13),(11,14),(12,15),(22,25),(23,26),(24,27),(28,31),(29,32),(30,33)], [(4,34),(5,35),(6,36),(10,13),(11,14),(12,15),(16,19),(17,20),(18,21),(28,31),(29,32),(30,33)], [(1,10,16),(2,11,17),(3,12,18),(4,25,31),(5,26,32),(6,27,33),(7,13,19),(8,14,20),(9,15,21),(22,28,34),(23,29,35),(24,30,36)], [(1,7),(2,9),(3,8),(4,28),(5,30),(6,29),(10,19),(11,21),(12,20),(13,16),(14,18),(15,17),(22,25),(23,27),(24,26),(31,34),(32,36),(33,35)])

Matrix representation ►G ⊆ 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	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	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	-1	-1	0
0	0	0	0	1	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	-1	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	2	0
0	0	0	0	0	-1	-1
0	0	0	0	0	1	0

0	1	0	0	0	0	0
1	0	0	0	0	0	0
0	0	0	1	0	0	0
0	0	1	0	0	0	0
0	0	0	0	-1	-2	0
0	0	0	0	0	1	0
0	0	0	0	0	-1	-1

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,-1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[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,-1,1,0,0,0,0,0,-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,-1,0,0,0,0,0,-1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,2,-1,1,0,0,0,0,0,-1,0],[0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-2,1,-1,0,0,0,0,0,0,-1] >;

36 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	2H	2I	2J	2K	3A	3B	3C	3D	4A	4B	4C	4D	6A	6B	6C	6D	6E	6F	6G	6H	···	6P
order	1	2	2	2	2	2	2	2	2	2	2	2	3	3	3	3	4	4	4	4	6	6	6	6	6	6	6	6	···	6
size	1	1	1	1	3	3	3	3	18	18	18	18	2	8	8	8	18	18	18	18	2	2	2	6	6	6	6	8	···	8

36 irreducible representations

dim	1	1	1	2	2	2	2	3	3	6	6
type	+	+	+	+	+	+	+	+	+	+	+
image	C₁	C₂	C₂	S₃	S₃	D₆	D₆	S₄	C₂×S₄	C₃⋊S₄	C₂×C₃⋊S₄
kernel	C₂²×C₃⋊S₄	C₂×C₃⋊S₄	A₄×C₂×C₆	C₂²×A₄	C₂³×C₆	C₂×A₄	C₂²×C₆	C₂×C₆	C₆	C₂²	C₂
# reps	1	6	1	3	1	9	3	2	6	1	3

In GAP, Magma, Sage, TeX

C_2^2\times C_3\rtimes S_4

% in TeX

G:=Group("C2^2xC3:S4");

// GroupNames label

G:=SmallGroup(288,1034);

// by ID

G=gap.SmallGroup(288,1034);

# by ID

G:=PCGroup([7,-2,-2,-2,-3,-3,-2,2,451,1684,6053,782,3534,1350]);

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^3=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,g*c*g=c^-1,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 = C₂²×C₃⋊S₄ order 288 = 2⁵·3²

Direct product of C₂² and C₃⋊S₄

G = C22×C3⋊S4 order 288 = 25·32

Direct product of C22 and C3⋊S4

G = C₂²×C₃⋊S₄ order 288 = 2⁵·3²

Direct product of C₂² and C₃⋊S₄