C6xC3.S4 - GroupNames

G = C₆×C₃.S₄ order 432 = 2⁴·3³

Direct product of C₆ and C₃.S₄

direct product, non-abelian, soluble, monomial

Aliases: C₆×C₃.S₄, C₆².₅₀D₆, C₂³⋊(C₃×D₉), C₂²⋊(C₆×D₉), (C₂×C₆)⋊₂D₁₈, C₃.₂(C₆×S₄), C₆.₁₁(C₃×S₄), (C₃×C₆).₁₇S₄, (C₂²×C₆)⋊₁D₉, C₃².₃(C₂×S₄), (C₂×C₆²).₁₅S₃, (C₂×C₃.A₄)⋊₄C₆, (C₆×C₃.A₄)⋊₂C₂, C₃.A₄⋊₄(C₂×C₆), (C₂×C₆).₃(S₃×C₆), (C₃×C₃.A₄)⋊₃C₂², (C₂²×C₆).₈(C₃×S₃), SmallGroup(432,534)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂² — C₃.A₄ — C₆×C₃.S₄

Chief series

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

Lower central

C₃.A₄ — C₆×C₃.S₄

Upper central

C₁ — C₆

Generators and relations for C₆×C₃.S₄
G = < a,b,c,d,e,f | a⁶=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: 622 in 124 conjugacy classes, 26 normal (22 characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₂², C₂², S₃, C₆, C₆, C₂×C₄, D₄, C₂³, C₂³, C₉, C₃², Dic₃, C₁₂, D₆, C₂×C₆, C₂×C₆, C₂×D₄, D₉, C₁₈, C₃×S₃, C₃×C₆, C₃×C₆, C₂×Dic₃, C₃⋊D₄, C₂×C₁₂, C₃×D₄, C₂²×S₃, C₂²×C₆, C₂²×C₆, C₃×C₉, C₃.A₄, C₃.A₄, D₁₈, C₃×Dic₃, S₃×C₆, C₆², C₆², C₂×C₃⋊D₄, C₆×D₄, C₃×D₉, C₃×C₁₈, C₃.S₄, C₂×C₃.A₄, C₂×C₃.A₄, C₆×Dic₃, C₃×C₃⋊D₄, S₃×C₂×C₆, C₂×C₆², C₃×C₃.A₄, C₆×D₉, C₂×C₃.S₄, C₆×C₃⋊D₄, C₃×C₃.S₄, C₆×C₃.A₄, C₆×C₃.S₄
Quotients: C₁, C₂, C₃, C₂², S₃, C₆, D₆, C₂×C₆, D₉, C₃×S₃, S₄, D₁₈, S₃×C₆, C₂×S₄, C₃×D₉, C₃.S₄, C₃×S₄, C₆×D₉, C₂×C₃.S₄, C₆×S₄, C₃×C₃.S₄, C₆×C₃.S₄

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

Generators in S₃₆

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

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

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

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

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

54 conjugacy classes

class	1	2A	2B	2C	2D	2E	3A	3B	3C	3D	3E	4A	4B	6A	6B	6C	6D	6E	6F	6G	6H	6I	6J	···	6O	6P	6Q	6R	6S	9A	···	9I	12A	12B	12C	12D	18A	···	18I
order	1	2	2	2	2	2	3	3	3	3	3	4	4	6	6	6	6	6	6	6	6	6	6	···	6	6	6	6	6	9	···	9	12	12	12	12	18	···	18
size	1	1	3	3	18	18	1	1	2	2	2	18	18	1	1	2	2	2	3	3	3	3	6	···	6	18	18	18	18	8	···	8	18	18	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	C₂	C₃	C₆	C₆	S₃	D₆	D₉	C₃×S₃	D₁₈	S₃×C₆	C₃×D₉	C₆×D₉	S₄	C₂×S₄	C₃×S₄	C₆×S₄	C₃.S₄	C₂×C₃.S₄	C₃×C₃.S₄	C₆×C₃.S₄
kernel	C₆×C₃.S₄	C₃×C₃.S₄	C₆×C₃.A₄	C₂×C₃.S₄	C₃.S₄	C₂×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	2	1	2	4	2	1	1	3	2	3	2	6	6	2	2	4	4	1	1	2	2

Matrix representation of C₆×C₃.S₄ ►in GL₇(𝔽₃₇)

36	0	0	0	0	0	0
0	36	0	0	0	0	0
0	0	11	0	0	0	0
0	0	0	11	0	0	0
0	0	0	0	36	0	0
0	0	0	0	0	36	0
0	0	0	0	0	0	36

26	0	0	0	0	0	0
33	10	0	0	0	0	0
0	0	1	0	0	0	0
0	0	0	1	0	0	0
0	0	0	0	1	0	0
0	0	0	0	0	1	0
0	0	0	0	0	0	1

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

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

16	0	0	0	0	0	0
7	7	0	0	0	0	0
0	0	0	1	0	0	0
0	0	36	36	0	0	0
0	0	0	0	0	0	1
0	0	0	0	1	0	0
0	0	0	0	0	1	0

30	9	0	0	0	0	0
7	7	0	0	0	0	0
0	0	26	20	0	0	0
0	0	31	11	0	0	0
0	0	0	0	0	0	36
0	0	0	0	0	36	0
0	0	0	0	36	0	0

G:=sub<GL(7,GF(37))| [36,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,36],[26,33,0,0,0,0,0,0,10,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,36],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,36],[16,7,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,36,0,0,0,0,0,1,36,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0],[30,7,0,0,0,0,0,9,7,0,0,0,0,0,0,0,26,31,0,0,0,0,0,20,11,0,0,0,0,0,0,0,0,0,36,0,0,0,0,0,36,0,0,0,0,0,36,0,0] >;

C₆×C₃.S₄ in GAP, Magma, Sage, TeX

C_6\times C_3.S_4

% in TeX

G:=Group("C6xC3.S4");

// GroupNames label

G:=SmallGroup(432,534);

// by ID

G=gap.SmallGroup(432,534);

# by ID

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

// Polycyclic

G:=Group<a,b,c,d,e,f|a^6=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 = C6×C3.S4 order 432 = 24·33

Direct product of C6 and C3.S4

G = C₆×C₃.S₄ order 432 = 2⁴·3³

Direct product of C₆ and C₃.S₄