C2^2xHe3:C2 - GroupNames

G = C₂²×He₃⋊C₂ order 216 = 2³·3³

Direct product of C₂² and He₃⋊C₂

direct product, non-abelian, supersoluble, monomial

Aliases: C₂²×He₃⋊C₂, C₆²⋊₇S₃, He₃⋊₃C₂³, (C₃×C₆)⋊₃D₆, (C₂×He₃)⋊₃C₂², (C₂²×He₃)⋊₅C₂, C₃²⋊₃(C₂²×S₃), C₆.₂₁(C₂×C₃⋊S₃), C₃.₂(C₂²×C₃⋊S₃), (C₂×C₆).₁₀(C₃⋊S₃), SmallGroup(216,113)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃ — He₃ — C₂²×He₃⋊C₂

Chief series

C₁ — C₃ — C₃² — He₃ — He₃⋊C₂ — C₂×He₃⋊C₂ — C₂²×He₃⋊C₂

Lower central

He₃ — C₂²×He₃⋊C₂

Upper central

C₁ — C₂×C₆

Generators and relations for C₂²×He₃⋊C₂
G = < a,b,c,d,e,f | a²=b²=c³=d³=e³=f²=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ece^-1=cd^-1, fcf=c^-1, de=ed, df=fd, fef=e^-1 >

Subgroups: 568 in 176 conjugacy classes, 46 normal (7 characteristic)
C₁, C₂, C₂, C₃, C₃, C₂², C₂², S₃, C₆, C₆, C₂³, C₃², D₆, C₂×C₆, C₂×C₆, C₃×S₃, C₃×C₆, C₂²×S₃, C₂²×C₆, He₃, S₃×C₆, C₆², He₃⋊C₂, C₂×He₃, S₃×C₂×C₆, C₂×He₃⋊C₂, C₂²×He₃, C₂²×He₃⋊C₂
Quotients: C₁, C₂, C₂², S₃, C₂³, D₆, C₃⋊S₃, C₂²×S₃, C₂×C₃⋊S₃, He₃⋊C₂, C₂²×C₃⋊S₃, C₂×He₃⋊C₂, C₂²×He₃⋊C₂

Smallest permutation representation of C₂²×He₃⋊C₂
►On 36 points

Generators in S₃₆

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

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

(2 15 11)(3 12 13)(4 8 36)(6 35 7)(17 24 20)(18 21 22)(26 33 29)(27 30 31)

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

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

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

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

C₂²×He₃⋊C₂ is a maximal subgroup of C₆².₅D₆ C₆².₃₁D₆ C₂²⋊(He₃⋊C₄) C₆²⋊₂D₆
C₂²×He₃⋊C₂ is a maximal quotient of C₆².₄₇D₆ C₆².₁₆D₆ He₃⋊₅D₄⋊C₂

40 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	3A	3B	3C	3D	3E	3F	6A	···	6F	6G	···	6R	6S	···	6Z
order	1	2	2	2	2	2	2	2	3	3	3	3	3	3	6	···	6	6	···	6	6	···	6
size	1	1	1	1	9	9	9	9	1	1	6	6	6	6	1	···	1	6	···	6	9	···	9

40 irreducible representations

dim	1	1	1	2	2	3	3
type	+	+	+	+	+
image	C₁	C₂	C₂	S₃	D₆	He₃⋊C₂	C₂×He₃⋊C₂
kernel	C₂²×He₃⋊C₂	C₂×He₃⋊C₂	C₂²×He₃	C₆²	C₃×C₆	C₂²	C₂
# reps	1	6	1	4	12	4	12

Matrix representation of C₂²×He₃⋊C₂ ►in GL₅(𝔽₇)

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

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

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

6	6	0	0	0
1	0	0	0	0
0	0	1	0	0
0	0	0	4	0
0	0	6	2	2

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

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

C₂²×He₃⋊C₂ in GAP, Magma, Sage, TeX

C_2^2\times {\rm He}_3\rtimes C_2

% in TeX

G:=Group("C2^2xHe3:C2");

// GroupNames label

G:=SmallGroup(216,113);

// by ID

G=gap.SmallGroup(216,113);

# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-3,387,1444,382]);

// Polycyclic

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

// generators/relations

G = C22×He3⋊C2 order 216 = 23·33

Direct product of C22 and He3⋊C2

G = C₂²×He₃⋊C₂ order 216 = 2³·3³

Direct product of C₂² and He₃⋊C₂