C3xC4:S4 - GroupNames

G = C₃×C₄⋊S₄ order 288 = 2⁵·3²

Direct product of C₃ and C₄⋊S₄

direct product, non-abelian, soluble, monomial

Aliases: C₃×C₄⋊S₄, C₁₂⋊₃S₄, C₄⋊(C₃×S₄), (C₆×S₄)⋊₄C₂, (C₂×S₄)⋊₁C₆, (C₄×A₄)⋊₁C₆, (C₃×A₄)⋊₄D₄, A₄⋊₁(C₃×D₄), (C₂×C₆)⋊₂D₁₂, C₂.₄(C₆×S₄), C₂²⋊(C₃×D₁₂), (C₁₂×A₄)⋊₂C₂, C₆.₄₁(C₂×S₄), (C₂²×C₁₂)⋊₄S₃, C₂³.₃(S₃×C₆), (C₂²×C₆).₁₀D₆, (C₆×A₄).₁₀C₂², (C₂×A₄).₃(C₂×C₆), (C₂²×C₄)⋊₂(C₃×S₃), SmallGroup(288,898)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂² — C₂×A₄ — C₃×C₄⋊S₄

Chief series

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

Lower central

A₄ — C₂×A₄ — C₃×C₄⋊S₄

Upper central

C₁ — C₆ — C₁₂

Generators and relations for C₃×C₄⋊S₄
G = < a,b,c,d,e,f | a³=b⁴=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: 466 in 118 conjugacy classes, 24 normal (20 characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₄, C₂², C₂², S₃, C₆, C₆, C₂×C₄, D₄, C₂³, C₂³, C₃², C₁₂, C₁₂, A₄, A₄, D₆, C₂×C₆, C₂×C₆, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂×D₄, C₃×S₃, C₃×C₆, D₁₂, C₂×C₁₂, C₃×D₄, S₄, C₂×A₄, C₂×A₄, C₂²×C₆, C₂²×C₆, C₄⋊D₄, C₃×C₁₂, C₃×A₄, S₃×C₆, C₃×C₂²⋊C₄, C₃×C₄⋊C₄, C₄×A₄, C₄×A₄, C₂²×C₁₂, C₆×D₄, C₂×S₄, C₃×D₁₂, C₃×S₄, C₆×A₄, C₃×C₄⋊D₄, C₄⋊S₄, C₁₂×A₄, C₆×S₄, C₃×C₄⋊S₄
Quotients: C₁, C₂, C₃, C₂², S₃, C₆, D₄, D₆, C₂×C₆, C₃×S₃, D₁₂, C₃×D₄, S₄, S₃×C₆, C₂×S₄, C₃×D₁₂, C₃×S₄, C₄⋊S₄, C₆×S₄, C₃×C₄⋊S₄

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

Generators in S₃₆

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

(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 3)(2 4)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)(25 27)(26 28)(33 35)(34 36)

(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(25 27)(26 28)(29 31)(30 32)

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

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

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

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

42 conjugacy classes

class	1	2A	2B	2C	2D	2E	3A	3B	3C	3D	3E	4A	4B	4C	4D	6A	6B	6C	6D	6E	6F	6G	6H	6I	6J	6K	6L	6M	12A	12B	12C	12D	12E	···	12J	12K	12L	12M	12N
order	1	2	2	2	2	2	3	3	3	3	3	4	4	4	4	6	6	6	6	6	6	6	6	6	6	6	6	6	12	12	12	12	12	···	12	12	12	12	12
size	1	1	3	3	12	12	1	1	8	8	8	2	6	12	12	1	1	3	3	3	3	8	8	8	12	12	12	12	2	2	6	6	8	···	8	12	12	12	12

42 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	2	2	3	3	3	3	6	6
type	+	+	+				+	+	+			+			+	+			+
image	C₁	C₂	C₂	C₃	C₆	C₆	S₃	D₄	D₆	C₃×S₃	C₃×D₄	D₁₂	S₃×C₆	C₃×D₁₂	S₄	C₂×S₄	C₃×S₄	C₆×S₄	C₄⋊S₄	C₃×C₄⋊S₄
kernel	C₃×C₄⋊S₄	C₁₂×A₄	C₆×S₄	C₄⋊S₄	C₄×A₄	C₂×S₄	C₂²×C₁₂	C₃×A₄	C₂²×C₆	C₂²×C₄	A₄	C₂×C₆	C₂³	C₂²	C₁₂	C₆	C₄	C₂	C₃	C₁
# reps	1	1	2	2	2	4	1	1	1	2	2	2	2	4	2	2	4	4	1	2

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

9	0	0	0	0
0	9	0	0	0
0	0	1	0	0
0	0	0	1	0
0	0	0	0	1

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

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

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

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

0	1	0	0	0
1	0	0	0	0
0	0	0	1	0
0	0	1	0	0
0	0	0	0	1

G:=sub<GL(5,GF(13))| [9,0,0,0,0,0,9,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[0,1,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,12,12,12,0,0,0,0,1,0,0,0,1,0],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,12,12,12],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,12,12,12,0,0,0,0,1],[0,1,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1] >;

C₃×C₄⋊S₄ in GAP, Magma, Sage, TeX

C_3\times C_4\rtimes S_4

% in TeX

G:=Group("C3xC4:S4");

// GroupNames label

G:=SmallGroup(288,898);

// by ID

G=gap.SmallGroup(288,898);

# by ID

G:=PCGroup([7,-2,-2,-3,-2,-3,-2,2,197,92,1684,6053,285,3534,475]);

// Polycyclic

G:=Group<a,b,c,d,e,f|a^3=b^4=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 = C3×C4⋊S4 order 288 = 25·32

Direct product of C3 and C4⋊S4

G = C₃×C₄⋊S₄ order 288 = 2⁵·3²

Direct product of C₃ and C₄⋊S₄