C2^2:(He3:C4) - GroupNames

G = C₂²⋊(He₃⋊C₄) order 432 = 2⁴·3³

The semidirect product of C₂² and He₃⋊C₄ acting via He₃⋊C₄/He₃⋊C₂=C₂

Aliases: C₂²⋊(He₃⋊C₄), C₃.(C₆²⋊C₄), He₃⋊₂(C₂²⋊C₄), He₃⋊C₂.₅D₄, (C₂²×He₃)⋊₁C₄, (C₂×He₃⋊C₄)⋊₂C₂, C₂.₇(C₂×He₃⋊C₄), C₆.₂₉(C₂×C₃²⋊C₄), (C₂×He₃⋊C₂)⋊₃C₄, (C₂×C₆).₂(C₃²⋊C₄), (C₂×He₃).₇(C₂×C₄), (C₂²×He₃⋊C₂).₃C₂, (C₂×He₃⋊C₂).₁₀C₂², SmallGroup(432,279)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃ — C₂×He₃ — C₂²⋊(He₃⋊C₄)

Chief series

C₁ — C₃ — He₃ — He₃⋊C₂ — C₂×He₃⋊C₂ — C₂×He₃⋊C₄ — C₂²⋊(He₃⋊C₄)

Lower central

He₃ — C₂×He₃ — C₂²⋊(He₃⋊C₄)

Upper central

C₁ — C₆ — C₂×C₆

Generators and relations for C₂²⋊(He₃⋊C₄)
G = < a,b,c,d,e,f | a²=b²=c³=d³=e³=f⁴=1, faf^-1=ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ece^-1=cd^-1, fcf^-1=cde, de=ed, df=fd, fef^-1=ce^-1 >

Subgroups: 701 in 115 conjugacy classes, 17 normal (13 characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₂², C₂², S₃, C₆, C₆, C₂×C₄, C₂³, C₃², C₁₂, D₆, C₂×C₆, C₂×C₆, C₂²⋊C₄, C₃×S₃, C₃×C₆, C₂×C₁₂, C₂²×S₃, C₂²×C₆, He₃, S₃×C₆, C₆², C₃×C₂²⋊C₄, He₃⋊C₂, He₃⋊C₂, C₂×He₃, C₂×He₃, S₃×C₂×C₆, He₃⋊C₄, C₂×He₃⋊C₂, C₂×He₃⋊C₂, C₂²×He₃, C₂×He₃⋊C₄, C₂²×He₃⋊C₂, C₂²⋊(He₃⋊C₄)
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, C₂²⋊C₄, C₃²⋊C₄, C₂×C₃²⋊C₄, He₃⋊C₄, C₆²⋊C₄, C₂×He₃⋊C₄, C₂²⋊(He₃⋊C₄)

Smallest permutation representation of C₂²⋊(He₃⋊C₄)
►On 36 points

Generators in S₃₆

(2 7)(4 11)(6 9)(14 17)(16 19)(21 33)(23 35)(26 29)(28 31)

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

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

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

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

(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)

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

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

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

38 conjugacy classes

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

38 irreducible representations

dim	1	1	1	1	1	2	3	3	4	4	4	6
type	+	+	+			+			+	+	+
image	C₁	C₂	C₂	C₄	C₄	D₄	He₃⋊C₄	C₂×He₃⋊C₄	C₃²⋊C₄	C₂×C₃²⋊C₄	C₆²⋊C₄	C₂²⋊(He₃⋊C₄)
kernel	C₂²⋊(He₃⋊C₄)	C₂×He₃⋊C₄	C₂²×He₃⋊C₂	C₂×He₃⋊C₂	C₂²×He₃	He₃⋊C₂	C₂²	C₂	C₂×C₆	C₆	C₃	C₁
# reps	1	2	1	2	2	2	8	8	2	2	4	4

Matrix representation of C₂²⋊(He₃⋊C₄) ►in GL₅(𝔽₁₃)

12	0	0	0	0
4	1	0	0	0
0	0	1	0	0
0	0	0	1	0
0	0	0	0	1

12	0	0	0	0
0	12	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	0	0	1
0	0	3	0	0
0	0	0	9	0

1	0	0	0	0
0	1	0	0	0
0	0	9	0	0
0	0	0	9	0
0	0	0	0	9

1	0	0	0	0
0	1	0	0	0
0	0	1	0	0
0	0	0	9	0
0	0	0	0	3

8	4	0	0	0
0	5	0	0	0
0	0	3	9	3
0	0	1	1	3
0	0	3	1	1

G:=sub<GL(5,GF(13))| [12,4,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[12,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,0,3,0,0,0,0,0,9,0,0,1,0,0],[1,0,0,0,0,0,1,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,9],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,9,0,0,0,0,0,3],[8,0,0,0,0,4,5,0,0,0,0,0,3,1,3,0,0,9,1,1,0,0,3,3,1] >;

C₂²⋊(He₃⋊C₄) in GAP, Magma, Sage, TeX

C_2^2\rtimes ({\rm He}_3\rtimes C_4)

% in TeX

G:=Group("C2^2:(He3:C4)");

// GroupNames label

G:=SmallGroup(432,279);

// by ID

G=gap.SmallGroup(432,279);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,28,141,3924,298,5381,2539,537]);

// Polycyclic

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

// generators/relations

G = C22⋊(He3⋊C4) order 432 = 24·33

The semidirect product of C22 and He3⋊C4 acting via He3⋊C4/He3⋊C2=C2

G = C₂²⋊(He₃⋊C₄) order 432 = 2⁴·3³

The semidirect product of C₂² and He₃⋊C₄ acting via He₃⋊C₄/He₃⋊C₂=C₂