C2xC6xS4 - GroupNames

G = C₂×C₆×S₄ order 288 = 2⁵·3²

Direct product of C₂×C₆ and S₄

direct product, non-abelian, soluble, monomial

Aliases: C₂×C₆×S₄, C₂³⋊(S₃×C₆), A₄⋊(C₂²×C₆), C₂⁴⋊₄(C₃×S₃), (C₂³×C₆)⋊₂S₃, (C₂²×C₆)⋊₂D₆, (C₆×A₄)⋊₂C₂², (C₂²×A₄)⋊₅C₆, (C₃×A₄)⋊₂C₂³, C₂²⋊(S₃×C₂×C₆), (C₂×A₄)⋊(C₂×C₆), (A₄×C₂×C₆)⋊₅C₂, (C₂×C₆)⋊₂(C₂²×S₃), SmallGroup(288,1033)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂² — A₄ — C₂×C₆×S₄

Chief series

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

Lower central

A₄ — C₂×C₆×S₄

Upper central

C₁ — C₂×C₆

Generators and relations for C₂×C₆×S₄
G = < a,b,c,d,e,f | a²=b⁶=c²=d²=e³=f²=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, ece^-1=fcf=cd=dc, ede^-1=c, df=fd, fef=e^-1 >

Subgroups: 890 in 272 conjugacy classes, 52 normal (14 characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₂², C₂², S₃, C₆, C₆, C₂×C₄, D₄, C₂³, C₂³, C₃², C₁₂, A₄, A₄, D₆, C₂×C₆, C₂×C₆, C₂²×C₄, C₂×D₄, C₂⁴, C₂⁴, C₃×S₃, C₃×C₆, C₂×C₁₂, C₃×D₄, S₄, C₂×A₄, C₂×A₄, C₂²×S₃, C₂²×C₆, C₂²×C₆, C₂²×D₄, C₃×A₄, S₃×C₆, C₆², C₂²×C₁₂, C₆×D₄, C₂×S₄, C₂²×A₄, C₂²×A₄, C₂³×C₆, C₂³×C₆, C₃×S₄, C₆×A₄, S₃×C₂×C₆, D₄×C₂×C₆, C₂²×S₄, C₆×S₄, A₄×C₂×C₆, C₂×C₆×S₄
Quotients: C₁, C₂, C₃, C₂², S₃, C₆, C₂³, D₆, C₂×C₆, C₃×S₃, S₄, C₂²×S₃, C₂²×C₆, S₃×C₆, C₂×S₄, C₃×S₄, S₃×C₂×C₆, C₂²×S₄, C₆×S₄, C₂×C₆×S₄

Smallest permutation representation of C₂×C₆×S₄
►On 36 points

Generators in S₃₆

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

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

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

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

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

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

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

60 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	2H	2I	2J	2K	3A	3B	3C	3D	3E	4A	4B	4C	4D	6A	···	6F	6G	···	6N	6O	···	6V	6W	···	6AE	12A	···	12H
order	1	2	2	2	2	2	2	2	2	2	2	2	3	3	3	3	3	4	4	4	4	6	···	6	6	···	6	6	···	6	6	···	6	12	···	12
size	1	1	1	1	3	3	3	3	6	6	6	6	1	1	8	8	8	6	6	6	6	1	···	1	3	···	3	6	···	6	8	···	8	6	···	6

60 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	3	3	3	3
type	+	+	+				+	+			+	+
image	C₁	C₂	C₂	C₃	C₆	C₆	S₃	D₆	C₃×S₃	S₃×C₆	S₄	C₂×S₄	C₃×S₄	C₆×S₄
kernel	C₂×C₆×S₄	C₆×S₄	A₄×C₂×C₆	C₂²×S₄	C₂×S₄	C₂²×A₄	C₂³×C₆	C₂²×C₆	C₂⁴	C₂³	C₂×C₆	C₆	C₂²	C₂
# reps	1	6	1	2	12	2	1	3	2	6	2	6	4	12

Matrix representation of C₂×C₆×S₄ ►in GL₅(𝔽₁₃)

12	0	0	0	0
0	12	0	0	0
0	0	12	0	0
0	0	0	12	0
0	0	0	0	12

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

1	0	0	0	0
0	1	0	0	0
0	0	12	0	0
0	0	0	12	0
0	0	1	1	1

1	0	0	0	0
0	1	0	0	0
0	0	1	0	0
0	0	0	12	0
0	0	12	0	12

0	12	0	0	0
1	12	0	0	0
0	0	12	12	11
0	0	1	0	0
0	0	0	0	1

12	1	0	0	0
0	1	0	0	0
0	0	12	0	0
0	0	1	1	2
0	0	0	0	12

G:=sub<GL(5,GF(13))| [12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[4,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,12,0,1,0,0,0,12,1,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,12,0,0,0,12,0,0,0,0,0,12],[0,1,0,0,0,12,12,0,0,0,0,0,12,1,0,0,0,12,0,0,0,0,11,0,1],[12,0,0,0,0,1,1,0,0,0,0,0,12,1,0,0,0,0,1,0,0,0,0,2,12] >;

C₂×C₆×S₄ in GAP, Magma, Sage, TeX

C_2\times C_6\times S_4

% in TeX

G:=Group("C2xC6xS4");

// GroupNames label

G:=SmallGroup(288,1033);

// by ID

G=gap.SmallGroup(288,1033);

# by ID

G:=PCGroup([7,-2,-2,-2,-3,-3,-2,2,1684,6053,285,3534,475]);

// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^6=c^2=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,e*c*e^-1=f*c*f=c*d=d*c,e*d*e^-1=c,d*f=f*d,f*e*f=e^-1>;

// generators/relations

G = C2×C6×S4 order 288 = 25·32

Direct product of C2×C6 and S4

G = C₂×C₆×S₄ order 288 = 2⁵·3²

Direct product of C₂×C₆ and S₄