C2^2xS3xA4

direct product, metabelian, soluble, monomial, A-group

Aliases: C₂²×S₃×A₄, C₆⋊(C₂²×A₄), C₃⋊(C₂³×A₄), (C₂³×C₆)⋊₅C₆, C₂³⋊₄(S₃×C₆), C₂⁴⋊₈(C₃×S₃), (C₆×A₄)⋊₄C₂², (S₃×C₂³)⋊₃C₆, (S₃×C₂⁴)⋊₁C₃, (C₃×A₄)⋊₄C₂³, (A₄×C₂×C₆)⋊₇C₂, (C₂×C₆)⋊₅(C₂×A₄), (C₂²×C₆)⋊(C₂×C₆), (C₂×C₆)⋊(C₂²×C₆), C₂²⋊₂(S₃×C₂×C₆), (C₂²×S₃)⋊₅(C₂×C₆), SmallGroup(288,1037)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₆ — C₂²×S₃×A₄

Chief series

C₁ — C₃ — C₂×C₆ — C₃×A₄ — S₃×A₄ — C₂×S₃×A₄ — C₂²×S₃×A₄

Lower central

C₂×C₆ — C₂²×S₃×A₄

Upper central

C₁ — C₂²

Subgroups: 1682 in 366 conjugacy classes, 63 normal (15 characteristic)
C₁, C₂^[×3], C₂^[×12], C₃, C₃^[×2], C₂²^[×2], C₂²^[×55], S₃^[×4], S₃^[×4], C₆^[×3], C₆^[×14], C₂³^[×3], C₂³^[×54], C₃², A₄, A₄, D₆^[×6], D₆^[×38], C₂×C₆^[×2], C₂×C₆^[×19], C₂⁴, C₂⁴^[×14], C₃×S₃^[×4], C₃×C₆^[×3], C₂×A₄^[×3], C₂×A₄^[×7], C₂²×S₃, C₂²×S₃^[×4], C₂²×S₃^[×45], C₂²×C₆^[×3], C₂²×C₆^[×5], C₂⁵, C₃×A₄, S₃×C₆^[×6], C₆², C₂²×A₄, C₂²×A₄^[×7], S₃×C₂³^[×6], S₃×C₂³^[×8], C₂³×C₆, S₃×A₄^[×4], C₆×A₄^[×3], S₃×C₂×C₆, C₂³×A₄, S₃×C₂⁴, C₂×S₃×A₄^[×6], A₄×C₂×C₆, C₂²×S₃×A₄

Quotients:
C₁, C₂^[×7], C₃, C₂²^[×7], S₃, C₆^[×7], C₂³, A₄, D₆^[×3], C₂×C₆^[×7], C₃×S₃, C₂×A₄^[×7], C₂²×S₃, C₂²×C₆, S₃×C₆^[×3], C₂²×A₄^[×7], S₃×A₄, S₃×C₂×C₆, C₂³×A₄, C₂×S₃×A₄^[×3], C₂²×S₃×A₄

Generators and relations
G = < a,b,c,d,e,f,g | a²=b²=c³=d²=e²=f²=g³=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, bd=db, be=eb, bf=fb, bg=gb, dcd=c^-1, ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, geg^-1=ef=fe, gfg^-1=e >

Smallest permutation representation
►On 36 points

Generators in S₃₆

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

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

(1 4)(2 5)(3 6)(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)(31 34)(32 35)(33 36)

(7 10)(8 11)(9 12)(13 16)(14 17)(15 18)(25 28)(26 29)(27 30)(31 34)(32 35)(33 36)

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

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

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

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

Matrix representation ►G ⊆ GL₇(𝔽₇)

6	0	0	0	0	0	0
0	6	0	0	0	0	0
0	0	6	0	0	0	0
0	0	0	6	0	0	0
0	0	0	0	1	0	0
0	0	0	0	0	1	0
0	0	0	0	0	0	1

6	0	0	0	0	0	0
0	6	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	6	0	0	0	0	0
1	6	0	0	0	0	0
0	0	6	6	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	6	0	0	0	0	0
0	6	0	0	0	0	0
0	0	6	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

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	6	0	0
0	0	0	0	0	1	0
0	0	0	0	0	0	6

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	6	0	0
0	0	0	0	0	6	0
0	0	0	0	0	0	1

2	0	0	0	0	0	0
0	2	0	0	0	0	0
0	0	1	0	0	0	0
0	0	0	1	0	0	0
0	0	0	0	0	0	1
0	0	0	0	6	0	0
0	0	0	0	0	6	0

G:=sub<GL(7,GF(7))| [6,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[6,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,6,6,0,0,0,0,0,0,0,6,1,0,0,0,0,0,6,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,6,6,0,0,0,0,0,0,0,6,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,6],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,1],[2,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,6,0,0,0,0,1,0,0] >;

48 conjugacy classes

class	1	2A	2B	2C	2D	···	2K	2L	2M	2N	2O	3A	3B	3C	3D	3E	6A	6B	6C	6D	···	6I	6J	6K	6L	6M	6N	···	6S	6T	···	6AA
order	1	2	2	2	2	···	2	2	2	2	2	3	3	3	3	3	6	6	6	6	···	6	6	6	6	6	6	···	6	6	···	6
size	1	1	1	1	3	···	3	9	9	9	9	2	4	4	8	8	2	2	2	4	···	4	6	6	6	6	8	···	8	12	···	12

48 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	3	3	3	6	6
type	+	+	+				+	+			+	+	+	+	+
image	C₁	C₂	C₂	C₃	C₆	C₆	S₃	D₆	C₃×S₃	S₃×C₆	A₄	C₂×A₄	C₂×A₄	S₃×A₄	C₂×S₃×A₄
kernel	C₂²×S₃×A₄	C₂×S₃×A₄	A₄×C₂×C₆	S₃×C₂⁴	S₃×C₂³	C₂³×C₆	C₂²×A₄	C₂×A₄	C₂⁴	C₂³	C₂²×S₃	D₆	C₂×C₆	C₂²	C₂
# reps	1	6	1	2	12	2	1	3	2	6	1	6	1	1	3

In GAP, Magma, Sage, TeX

C_2^2\times S_3\times A_4

% in TeX

G:=Group("C2^2xS3xA4");

// GroupNames label

G:=SmallGroup(288,1037);

// by ID

G=gap.SmallGroup(288,1037);

# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,2,-3,340,152,9414]);

// Polycyclic

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

// generators/relations

G = C₂²×S₃×A₄ order 288 = 2⁵·3²

Direct product of C₂², S₃ and A₄

G = C22×S3×A4 order 288 = 25·32

Direct product of C22, S3 and A4

G = C₂²×S₃×A₄ order 288 = 2⁵·3²

Direct product of C₂², S₃ and A₄