C2xS3xHe3 - GroupNames

G = C₂×S₃×He₃ order 324 = 2²·3⁴

Direct product of C₂, S₃ and He₃

direct product, metabelian, supersoluble, monomial

Aliases: C₂×S₃×He₃, C₃².₈C₆², C₆⋊(C₂×He₃), C₃⋊(C₂²×He₃), (C₆×He₃)⋊₂C₂, C₃²⋊₈(S₃×C₆), C₃³⋊₄(C₂×C₆), (C₃²×C₆)⋊₃C₆, (S₃×C₃²)⋊₂C₆, (C₃×He₃)⋊₇C₂², (S₃×C₆).₂C₃², C₆.₁₂(S₃×C₃²), (S₃×C₃×C₆)⋊C₃, C₃.₅(S₃×C₃×C₆), (C₃×C₆)⋊₄(C₃×S₃), (C₃×S₃).₂(C₃×C₆), (C₃×C₆).₁₉(C₃×C₆), SmallGroup(324,139)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃² — C₂×S₃×He₃

Chief series

C₁ — C₃ — C₃² — C₃³ — C₃×He₃ — S₃×He₃ — C₂×S₃×He₃

Lower central

C₃ — C₃² — C₂×S₃×He₃

Upper central

C₁ — C₆ — C₂×He₃

Generators and relations for C₂×S₃×He₃
G = < a,b,c,d,e,f | a²=b³=c²=d³=e³=f³=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, cbc=b^-1, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, fdf^-1=de^-1, ef=fe >

Subgroups: 436 in 144 conjugacy classes, 49 normal (15 characteristic)
C₁, C₂, C₂, C₃, C₃, C₂², S₃, C₆, C₆, C₃², C₃², C₃², D₆, C₂×C₆, C₃×S₃, C₃×S₃, C₃×C₆, C₃×C₆, C₃×C₆, He₃, He₃, C₃³, S₃×C₆, S₃×C₆, C₆², C₂×He₃, C₂×He₃, S₃×C₃², C₃²×C₆, C₃×He₃, C₂²×He₃, S₃×C₃×C₆, S₃×He₃, C₆×He₃, C₂×S₃×He₃
Quotients: C₁, C₂, C₃, C₂², S₃, C₆, C₃², D₆, C₂×C₆, C₃×S₃, C₃×C₆, He₃, S₃×C₆, C₆², C₂×He₃, S₃×C₃², C₂²×He₃, S₃×C₃×C₆, S₃×He₃, C₂×S₃×He₃

Smallest permutation representation of C₂×S₃×He₃
►On 36 points

Generators in S₃₆

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

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

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

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

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

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

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

66 conjugacy classes

class	1	2A	2B	2C	3A	3B	3C	3D	3E	3F	···	3M	3N	···	3U	6A	6B	6C	6D	6E	6F	···	6Q	6R	···	6Y	6Z	···	6AO
order	1	2	2	2	3	3	3	3	3	3	···	3	3	···	3	6	6	6	6	6	6	···	6	6	···	6	6	···	6
size	1	1	3	3	1	1	2	2	2	3	···	3	6	···	6	1	1	2	2	2	3	···	3	6	···	6	9	···	9

66 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	3	3	3	6	6
type	+	+	+				+	+
image	C₁	C₂	C₂	C₃	C₆	C₆	S₃	D₆	C₃×S₃	S₃×C₆	He₃	C₂×He₃	C₂×He₃	S₃×He₃	C₂×S₃×He₃
kernel	C₂×S₃×He₃	S₃×He₃	C₆×He₃	S₃×C₃×C₆	S₃×C₃²	C₃²×C₆	C₂×He₃	He₃	C₃×C₆	C₃²	D₆	S₃	C₆	C₂	C₁
# reps	1	2	1	8	16	8	1	1	8	8	2	4	2	2	2

Matrix representation of C₂×S₃×He₃ ►in GL₅(𝔽₇)

6	0	0	0	0
0	6	0	0	0
0	0	1	0	0
0	0	0	1	0
0	0	0	0	1

0	1	0	0	0
6	6	0	0	0
0	0	1	0	0
0	0	0	1	0
0	0	0	0	1

0	6	0	0	0
6	0	0	0	0
0	0	1	0	0
0	0	0	1	0
0	0	0	0	1

2	0	0	0	0
0	2	0	0	0
0	0	1	5	6
0	0	0	0	6
0	0	0	1	6

1	0	0	0	0
0	1	0	0	0
0	0	4	0	0
0	0	0	4	0
0	0	0	0	4

1	0	0	0	0
0	1	0	0	0
0	0	0	0	6
0	0	3	1	6
0	0	1	0	6

G:=sub<GL(5,GF(7))| [6,0,0,0,0,0,6,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[0,6,0,0,0,1,6,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[0,6,0,0,0,6,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[2,0,0,0,0,0,2,0,0,0,0,0,1,0,0,0,0,5,0,1,0,0,6,6,6],[1,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4],[1,0,0,0,0,0,1,0,0,0,0,0,0,3,1,0,0,0,1,0,0,0,6,6,6] >;

C₂×S₃×He₃ in GAP, Magma, Sage, TeX

C_2\times S_3\times {\rm He}_3

% in TeX

G:=Group("C2xS3xHe3");

// GroupNames label

G:=SmallGroup(324,139);

// by ID

G=gap.SmallGroup(324,139);

# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-3,303,7781]);

// Polycyclic

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

// generators/relations

G = C2×S3×He3 order 324 = 22·34

Direct product of C2, S3 and He3

G = C₂×S₃×He₃ order 324 = 2²·3⁴

Direct product of C₂, S₃ and He₃