C2xHe3:5S3 - GroupNames

G = C₂×He₃⋊₅S₃ order 324 = 2²·3⁴

Direct product of C₂ and He₃⋊₅S₃

direct product, non-abelian, supersoluble, monomial

Aliases: C₂×He₃⋊₅S₃, He₃⋊₁₀D₆, C₃³⋊₁₂D₆, C₆⋊(He₃⋊C₂), (C₂×He₃)⋊₅S₃, (C₆×He₃)⋊₅C₂, (C₃²×C₆)⋊₆S₃, (C₃×He₃)⋊₁₀C₂², C₆.₅(C₃³⋊C₂), (C₃×C₆)⋊₂(C₃⋊S₃), C₃²⋊₃(C₂×C₃⋊S₃), C₃⋊₂(C₂×He₃⋊C₂), C₃.₂(C₂×C₃³⋊C₂), SmallGroup(324,150)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃ — C₃×He₃ — C₂×He₃⋊₅S₃

Chief series

C₁ — C₃ — C₃² — He₃ — C₃×He₃ — He₃⋊₅S₃ — C₂×He₃⋊₅S₃

Lower central

C₃×He₃ — C₂×He₃⋊₅S₃

Upper central

C₁ — C₆

Generators and relations for C₂×He₃⋊₅S₃
G = < a,b,c,d,e,f | a²=b³=c³=d³=e³=f²=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ebe^-1=bc=cb, dbd^-1=bc^-1, fbf=b^-1, cd=dc, ce=ec, cf=fc, de=ed, fdf=d^-1, fef=e^-1 >

Subgroups: 1228 in 240 conjugacy classes, 63 normal (11 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, C₃³, S₃×C₆, C₂×C₃⋊S₃, He₃⋊C₂, C₂×He₃, C₃×C₃⋊S₃, C₃²×C₆, C₃×He₃, C₂×He₃⋊C₂, C₆×C₃⋊S₃, He₃⋊₅S₃, C₆×He₃, C₂×He₃⋊₅S₃
Quotients: C₁, C₂, C₂², S₃, D₆, C₃⋊S₃, C₂×C₃⋊S₃, He₃⋊C₂, C₃³⋊C₂, C₂×He₃⋊C₂, C₂×C₃³⋊C₂, He₃⋊₅S₃, C₂×He₃⋊₅S₃

Smallest permutation representation of C₂×He₃⋊₅S₃
►On 36 points

Generators in S₃₆

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

(1 10 14)(3 13 12)(4 36 9)(6 8 35)(16 22 20)(18 19 24)(26 32 28)(27 29 33)

(2 11 15)(3 13 12)(4 36 9)(5 7 34)(17 23 21)(18 19 24)(25 31 30)(27 29 33)

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

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

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

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

42 conjugacy classes

class	1	2A	2B	2C	3A	3B	3C	3D	3E	3F	···	3Q	6A	6B	6C	6D	6E	6F	···	6Q	6R	6S	6T	6U
order	1	2	2	2	3	3	3	3	3	3	···	3	6	6	6	6	6	6	···	6	6	6	6	6
size	1	1	27	27	1	1	2	2	2	6	···	6	1	1	2	2	2	6	···	6	27	27	27	27

42 irreducible representations

dim	1	1	1	2	2	2	2	3	3	6	6
type	+	+	+	+	+	+	+
image	C₁	C₂	C₂	S₃	S₃	D₆	D₆	He₃⋊C₂	C₂×He₃⋊C₂	He₃⋊₅S₃	C₂×He₃⋊₅S₃
kernel	C₂×He₃⋊₅S₃	He₃⋊₅S₃	C₆×He₃	C₂×He₃	C₃²×C₆	He₃	C₃³	C₆	C₃	C₂	C₁
# reps	1	2	1	9	4	9	4	4	4	2	2

Matrix representation of C₂×He₃⋊₅S₃ ►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

1	0	0	0	0
0	1	0	0	0
0	0	1	0	0
0	0	2	2	2
0	0	0	0	4

1	0	0	0	0
0	1	0	0	0
0	0	2	0	0
0	0	0	2	0
0	0	0	0	2

1	0	0	0	0
0	1	0	0	0
0	0	6	3	4
0	0	0	1	0
0	0	5	0	0

0	6	0	0	0
1	6	0	0	0
0	0	0	0	3
0	0	0	1	0
0	0	2	1	6

0	1	0	0	0
1	0	0	0	0
0	0	1	0	0
0	0	0	1	0
0	0	2	1	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],[1,0,0,0,0,0,1,0,0,0,0,0,1,2,0,0,0,0,2,0,0,0,0,2,4],[1,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,0,2,0,0,0,0,0,2],[1,0,0,0,0,0,1,0,0,0,0,0,6,0,5,0,0,3,1,0,0,0,4,0,0],[0,1,0,0,0,6,6,0,0,0,0,0,0,0,2,0,0,0,1,1,0,0,3,0,6],[0,1,0,0,0,1,0,0,0,0,0,0,1,0,2,0,0,0,1,1,0,0,0,0,6] >;

C₂×He₃⋊₅S₃ in GAP, Magma, Sage, TeX

C_2\times {\rm He}_3\rtimes_5S_3

% in TeX

G:=Group("C2xHe3:5S3");

// GroupNames label

G:=SmallGroup(324,150);

// by ID

G=gap.SmallGroup(324,150);

# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-3,146,579,2164,382]);

// Polycyclic

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

// generators/relations

G = C2×He3⋊5S3 order 324 = 22·34

Direct product of C2 and He3⋊5S3

G = C₂×He₃⋊₅S₃ order 324 = 2²·3⁴

Direct product of C₂ and He₃⋊₅S₃