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⋊₂S₄, C₃⋊₂(C₄×S₄), (C₄×A₄)⋊₂S₃, A₄⋊₂(C₄×S₃), (C₁₂×A₄)⋊₅C₂, C₆.₂₉(C₂×S₄), (C₂×A₄).₉D₆, C₆.₇S₄⋊₅C₂, (C₂²×C₁₂)⋊₃S₃, (C₂²×C₆).₂₀D₆, (C₆×A₄).₁₄C₂², C₂²⋊(C₄×C₃⋊S₃), (C₂×C₆)⋊₄(C₄×S₃), C₂.₁(C₂×C₃⋊S₄), (C₃×A₄)⋊₅(C₂×C₄), (C₂×C₃⋊S₄).₂C₂, C₂³.₂(C₂×C₃⋊S₃), (C₂²×C₄)⋊₁(C₃⋊S₃), SmallGroup(288,908)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂² — C₃×A₄ — C₄×C₃⋊S₄

Chief series

C₁ — C₂² — C₂×C₆ — C₃×A₄ — C₆×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=e³=f²=1, 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=e^-1 >

Subgroups: 784 in 144 conjugacy classes, 29 normal (18 characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₄, C₂², C₂², S₃, C₆, C₆, C₂×C₄, D₄, C₂³, C₂³, C₃², Dic₃, C₁₂, C₁₂, A₄, D₆, C₂×C₆, C₂×C₆, C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₃⋊S₃, C₃×C₆, C₄×S₃, C₂×Dic₃, C₃⋊D₄, C₂×C₁₂, S₄, C₂×A₄, C₂²×S₃, C₂²×C₆, C₄×D₄, C₃⋊Dic₃, C₃×C₁₂, C₃×A₄, C₂×C₃⋊S₃, C₄×Dic₃, Dic₃⋊C₄, D₆⋊C₄, C₆.D₄, A₄⋊C₄, C₄×A₄, S₃×C₂×C₄, C₂×C₃⋊D₄, C₂²×C₁₂, C₂×S₄, C₄×C₃⋊S₃, C₃⋊S₄, C₆×A₄, C₄×C₃⋊D₄, C₄×S₄, C₆.₇S₄, C₁₂×A₄, C₂×C₃⋊S₄, C₄×C₃⋊S₄
Quotients: C₁, C₂, C₄, C₂², S₃, C₂×C₄, D₆, C₃⋊S₃, C₄×S₃, S₄, C₂×C₃⋊S₃, C₂×S₄, C₄×C₃⋊S₃, C₃⋊S₄, C₄×S₄, C₂×C₃⋊S₄, C₄×C₃⋊S₄

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

Generators in S₃₆

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

(1 3)(2 4)(5 7)(6 8)(13 15)(14 16)(21 23)(22 24)(25 27)(26 28)(29 31)(30 32)

(9 11)(10 12)(17 19)(18 20)(21 23)(22 24)(25 27)(26 28)(29 31)(30 32)(33 35)(34 36)

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

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

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

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

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

36 conjugacy classes

class	1	2A	2B	2C	2D	2E	3A	3B	3C	3D	4A	4B	4C	4D	4E	···	4J	6A	6B	6C	6D	6E	6F	12A	12B	12C	12D	12E	···	12J
order	1	2	2	2	2	2	3	3	3	3	4	4	4	4	4	···	4	6	6	6	6	6	6	12	12	12	12	12	···	12
size	1	1	3	3	18	18	2	8	8	8	1	1	3	3	18	···	18	2	6	6	8	8	8	2	2	6	6	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	S₃	D₆	D₆	C₄×S₃	C₄×S₃	S₄	C₂×S₄	C₄×S₄	C₃⋊S₄	C₂×C₃⋊S₄	C₄×C₃⋊S₄
kernel	C₄×C₃⋊S₄	C₆.₇S₄	C₁₂×A₄	C₂×C₃⋊S₄	C₃⋊S₄	C₄×A₄	C₂²×C₁₂	C₂×A₄	C₂²×C₆	A₄	C₂×C₆	C₁₂	C₆	C₃	C₄	C₂	C₁
# reps	1	1	1	1	4	3	1	3	1	6	2	2	2	4	1	1	2

Matrix representation of C₄×C₃⋊S₄ ►in GL₅(𝔽₁₃)

8	0	0	0	0
0	8	0	0	0
0	0	12	0	0
0	0	0	12	0
0	0	0	0	12

11	2	0	0	0
5	1	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	1	0	0
0	0	12	12	0
0	0	1	0	12

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

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

2	11	0	0	0
8	11	0	0	0
0	0	12	11	0
0	0	0	1	0
0	0	0	12	12

G:=sub<GL(5,GF(13))| [8,0,0,0,0,0,8,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[11,5,0,0,0,2,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,12,1,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,12,0,12,0,0,0,12,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,2,12,1,0,0,0,12,0],[2,8,0,0,0,11,11,0,0,0,0,0,12,0,0,0,0,11,1,12,0,0,0,0,12] >;

C₄×C₃⋊S₄ in GAP, Magma, Sage, TeX

C_4\times C_3\rtimes S_4

% in TeX

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

// GroupNames label

G:=SmallGroup(288,908);

// by ID

G=gap.SmallGroup(288,908);

# by ID

G:=PCGroup([7,-2,-2,-2,-3,-3,-2,2,36,451,1684,6053,782,3534,1350]);

// Polycyclic

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

// 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