C3xC6.S4 - GroupNames

G = C₃×C₆.S₄ order 432 = 2⁴·3³

Direct product of C₃ and C₆.S₄

direct product, non-abelian, soluble, monomial

Aliases: C₃×C₆.S₄, C₆².₉Dic₃, C₆.₈(C₃×S₄), C₂³.(C₃×D₉), C₃.A₄⋊₄C₁₂, (C₃×C₆).₁₃S₄, (C₂×C₆)⋊₁Dic₉, C₂²⋊(C₃×Dic₉), (C₂×C₆²).₆S₃, (C₂²×C₆).₁D₉, C₆.₁₀(C₃.S₄), C₃².₂(A₄⋊C₄), C₃.₂(C₃×A₄⋊C₄), (C₃×C₃.A₄)⋊₂C₄, C₂.₁(C₃×C₃.S₄), (C₂×C₃.A₄).₄C₆, (C₆×C₃.A₄).₂C₂, (C₂²×C₆).₅(C₃×S₃), (C₂×C₆).₃(C₃×Dic₃), SmallGroup(432,250)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂² — C₃.A₄ — C₃×C₆.S₄

Chief series

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

Lower central

C₃.A₄ — C₃×C₆.S₄

Upper central

C₁ — C₆

Generators and relations for C₃×C₆.S₄
G = < a,b,c,d,e,f | a³=b⁶=c²=d²=1, e³=b², f²=b³, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf^-1=b^-1, ece^-1=fcf^-1=cd=dc, ede^-1=c, df=fd, fef^-1=b⁴e² >

Subgroups: 314 in 76 conjugacy classes, 22 normal (all characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₂², C₂², C₆, C₆, C₂×C₄, C₂³, C₉, C₃², Dic₃, C₁₂, C₂×C₆, C₂×C₆, C₂²⋊C₄, C₁₈, C₃×C₆, C₃×C₆, C₂×Dic₃, C₂×C₁₂, C₂²×C₆, C₂²×C₆, C₃×C₉, Dic₉, C₃.A₄, C₃.A₄, C₃×Dic₃, C₆², C₆², C₆.D₄, C₃×C₂²⋊C₄, C₃×C₁₈, C₂×C₃.A₄, C₂×C₃.A₄, C₆×Dic₃, C₂×C₆², C₃×Dic₉, C₃×C₃.A₄, C₆.S₄, C₃×C₆.D₄, C₆×C₃.A₄, C₃×C₆.S₄
Quotients: C₁, C₂, C₃, C₄, S₃, C₆, Dic₃, C₁₂, D₉, C₃×S₃, S₄, Dic₉, C₃×Dic₃, A₄⋊C₄, C₃×D₉, C₃.S₄, C₃×S₄, C₃×Dic₉, C₆.S₄, C₃×A₄⋊C₄, C₃×C₃.S₄, C₃×C₆.S₄

Smallest permutation representation of C₃×C₆.S₄
►On 36 points

Generators in S₃₆

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

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

(1 27)(3 20)(4 21)(6 23)(7 24)(9 26)(11 30)(12 31)(14 33)(15 34)(17 36)(18 28)

(1 27)(2 19)(4 21)(5 22)(7 24)(8 25)(10 29)(12 31)(13 32)(15 34)(16 35)(18 28)

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

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

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

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

54 conjugacy classes

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

54 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	2	2	3	3	3	3	6	6	6	6
type	+	+					+	-	+		-				+				+	-
image	C₁	C₂	C₃	C₄	C₆	C₁₂	S₃	Dic₃	D₉	C₃×S₃	Dic₉	C₃×Dic₃	C₃×D₉	C₃×Dic₉	S₄	A₄⋊C₄	C₃×S₄	C₃×A₄⋊C₄	C₃.S₄	C₆.S₄	C₃×C₃.S₄	C₃×C₆.S₄
kernel	C₃×C₆.S₄	C₆×C₃.A₄	C₆.S₄	C₃×C₃.A₄	C₂×C₃.A₄	C₃.A₄	C₂×C₆²	C₆²	C₂²×C₆	C₂²×C₆	C₂×C₆	C₂×C₆	C₂³	C₂²	C₃×C₆	C₃²	C₆	C₃	C₆	C₃	C₂	C₁
# reps	1	1	2	2	2	4	1	1	3	2	3	2	6	6	2	2	4	4	1	1	2	2

Matrix representation of C₃×C₆.S₄ ►in GL₅(𝔽₃₇)

1	0	0	0	0
0	1	0	0	0
0	0	26	0	0
0	0	0	26	0
0	0	0	0	26

11	0	0	0	0
0	27	0	0	0
0	0	36	0	0
0	0	0	36	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	17	0	1

1	0	0	0	0
0	1	0	0	0
0	0	36	0	0
0	0	21	1	0
0	0	0	0	36

33	0	0	0	0
0	9	0	0	0
0	0	16	35	0
0	0	7	21	1
0	0	30	17	0

0	6	0	0	0
6	0	0	0	0
0	0	9	0	25
0	0	24	6	15
0	0	13	0	28

G:=sub<GL(5,GF(37))| [1,0,0,0,0,0,1,0,0,0,0,0,26,0,0,0,0,0,26,0,0,0,0,0,26],[11,0,0,0,0,0,27,0,0,0,0,0,36,0,0,0,0,0,36,0,0,0,0,0,36],[1,0,0,0,0,0,1,0,0,0,0,0,36,0,17,0,0,0,36,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,36,21,0,0,0,0,1,0,0,0,0,0,36],[33,0,0,0,0,0,9,0,0,0,0,0,16,7,30,0,0,35,21,17,0,0,0,1,0],[0,6,0,0,0,6,0,0,0,0,0,0,9,24,13,0,0,0,6,0,0,0,25,15,28] >;

C₃×C₆.S₄ in GAP, Magma, Sage, TeX

C_3\times C_6.S_4

% in TeX

G:=Group("C3xC6.S4");

// GroupNames label

G:=SmallGroup(432,250);

// by ID

G=gap.SmallGroup(432,250);

# by ID

G:=PCGroup([7,-2,-3,-2,-3,-3,-2,2,42,1683,192,2524,9077,782,5298,1350]);

// Polycyclic

G:=Group<a,b,c,d,e,f|a^3=b^6=c^2=d^2=1,e^3=b^2,f^2=b^3,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^-1=b^-1,e*c*e^-1=f*c*f^-1=c*d=d*c,e*d*e^-1=c,d*f=f*d,f*e*f^-1=b^4*e^2>;

// generators/relations

G = C3×C6.S4 order 432 = 24·33

Direct product of C3 and C6.S4

G = C₃×C₆.S₄ order 432 = 2⁴·3³

Direct product of C₃ and C₆.S₄