C4xC3.S4 - GroupNames

G = C₄×C₃.S₄ order 288 = 2⁵·3²

Direct product of C₄ and C₃.S₄

direct product, non-abelian, soluble, monomial

Aliases: C₄×C₃.S₄, C₁₂.₁₀S₄, C₂³.₂D₁₈, C₃.(C₄×S₄), C₂²⋊(C₄×D₉), C₆.₁₇(C₂×S₄), (C₂²×C₄)⋊₁D₉, C₆.S₄⋊₂C₂, (C₂²×C₁₂).₆S₃, (C₂²×C₆).₁₄D₆, (C₂×C₆).(C₄×S₃), (C₂×C₃.S₄).C₂, (C₄×C₃.A₄)⋊₂C₂, C₃.A₄⋊₁(C₂×C₄), C₂.₁(C₂×C₃.S₄), (C₂×C₃.A₄).₂C₂², SmallGroup(288,333)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂² — C₃.A₄ — C₄×C₃.S₄

Chief series

C₁ — C₂² — C₂×C₆ — C₃.A₄ — C₂×C₃.A₄ — C₂×C₃.S₄ — C₄×C₃.S₄

Lower central

C₃.A₄ — C₄×C₃.S₄

Upper central

C₁ — C₄

Generators and relations for C₄×C₃.S₄
G = < a,b,c,d,e,f | a⁴=b³=c²=d²=f²=1, e³=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf=b^-1, ece^-1=fcf=cd=dc, ede^-1=c, df=fd, fef=b^-1e² >

Subgroups: 514 in 96 conjugacy classes, 20 normal (18 characteristic)
C₁, C₂, C₂^[×4], C₃, C₄, C₄^[×5], C₂², C₂²^[×6], S₃^[×2], C₆, C₆^[×2], C₂×C₄^[×7], D₄^[×4], C₂³, C₂³, C₉, Dic₃^[×4], C₁₂, C₁₂, D₆^[×4], C₂×C₆, C₂×C₆^[×2], C₄², C₂²⋊C₄^[×2], C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, D₉^[×2], C₁₈, C₄×S₃^[×2], C₂×Dic₃^[×3], C₃⋊D₄^[×4], C₂×C₁₂^[×2], C₂²×S₃, C₂²×C₆, C₄×D₄, Dic₉, C₃₆, C₃.A₄, D₁₈, C₄×Dic₃, Dic₃⋊C₄, D₆⋊C₄, C₆.D₄, S₃×C₂×C₄, C₂×C₃⋊D₄, C₂²×C₁₂, C₄×D₉, C₃.S₄^[×2], C₂×C₃.A₄, C₄×C₃⋊D₄, C₆.S₄, C₄×C₃.A₄, C₂×C₃.S₄, C₄×C₃.S₄
Quotients: C₁, C₂^[×3], C₄^[×2], C₂², S₃, C₂×C₄, D₆, D₉, C₄×S₃, S₄, D₁₈, C₂×S₄, C₄×D₉, C₃.S₄, C₄×S₄, C₂×C₃.S₄, C₄×C₃.S₄

Smallest permutation representation of C₄×C₃.S₄
►On 36 points

Generators in S₃₆

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

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

(1 22)(2 23)(4 25)(5 26)(7 19)(8 20)(11 34)(12 35)(14 28)(15 29)(17 31)(18 32)

(2 23)(3 24)(5 26)(6 27)(8 20)(9 21)(10 33)(12 35)(13 36)(15 29)(16 30)(18 32)

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

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

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

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

36 conjugacy classes

class	1	2A	2B	2C	2D	2E	3	4A	4B	4C	4D	4E	···	4J	6A	6B	6C	9A	9B	9C	12A	12B	12C	12D	18A	18B	18C	36A	···	36F
order	1	2	2	2	2	2	3	4	4	4	4	4	···	4	6	6	6	9	9	9	12	12	12	12	18	18	18	36	···	36
size	1	1	3	3	18	18	2	1	1	3	3	18	···	18	2	6	6	8	8	8	2	2	6	6	8	8	8	8	···	8

36 irreducible representations

dim	1	1	1	1	1	2	2	2	2	2	2	3	3	3	6	6	6
type	+	+	+	+		+	+	+		+		+	+		+	+
image	C₁	C₂	C₂	C₂	C₄	S₃	D₆	D₉	C₄×S₃	D₁₈	C₄×D₉	S₄	C₂×S₄	C₄×S₄	C₃.S₄	C₂×C₃.S₄	C₄×C₃.S₄
kernel	C₄×C₃.S₄	C₆.S₄	C₄×C₃.A₄	C₂×C₃.S₄	C₃.S₄	C₂²×C₁₂	C₂²×C₆	C₂²×C₄	C₂×C₆	C₂³	C₂²	C₁₂	C₆	C₃	C₄	C₂	C₁
# reps	1	1	1	1	4	1	1	3	2	3	6	2	2	4	1	1	2

Matrix representation of C₄×C₃.S₄ ►in GL₅(𝔽₃₇)

6	0	0	0	0
0	6	0	0	0
0	0	36	0	0
0	0	0	36	0
0	0	0	0	36

36	36	0	0	0
1	0	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	36	0	0
0	0	1	1	0
0	0	0	0	36

1	0	0	0	0
0	1	0	0	0
0	0	36	0	0
0	0	0	36	0
0	0	36	0	1

26	6	0	0	0
31	20	0	0	0
0	0	1	0	35
0	0	0	0	1
0	0	0	36	36

1	0	0	0	0
36	36	0	0	0
0	0	36	35	0
0	0	0	1	0
0	0	0	36	36

G:=sub<GL(5,GF(37))| [6,0,0,0,0,0,6,0,0,0,0,0,36,0,0,0,0,0,36,0,0,0,0,0,36],[36,1,0,0,0,36,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,36,1,0,0,0,0,1,0,0,0,0,0,36],[1,0,0,0,0,0,1,0,0,0,0,0,36,0,36,0,0,0,36,0,0,0,0,0,1],[26,31,0,0,0,6,20,0,0,0,0,0,1,0,0,0,0,0,0,36,0,0,35,1,36],[1,36,0,0,0,0,36,0,0,0,0,0,36,0,0,0,0,35,1,36,0,0,0,0,36] >;

C₄×C₃.S₄ in GAP, Magma, Sage, TeX

C_4\times C_3.S_4

% in TeX

G:=Group("C4xC3.S4");

// GroupNames label

G:=SmallGroup(288,333);

// by ID

G=gap.SmallGroup(288,333);

# by ID

G:=PCGroup([7,-2,-2,-2,-3,-3,-2,2,36,1123,192,1684,6053,782,3534,1350]);

// Polycyclic

G:=Group<a,b,c,d,e,f|a^4=b^3=c^2=d^2=f^2=1,e^3=b,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,f*b*f=b^-1,e*c*e^-1=f*c*f=c*d=d*c,e*d*e^-1=c,d*f=f*d,f*e*f=b^-1*e^2>;

// generators/relations

G = C4×C3.S4 order 288 = 25·32

Direct product of C4 and C3.S4

G = C₄×C₃.S₄ order 288 = 2⁵·3²

Direct product of C₄ and C₃.S₄