C2^2xC3^2:A4 - GroupNames

G = C₂²×C₃²⋊A₄ order 432 = 2⁴·3³

Direct product of C₂² and C₃²⋊A₄

direct product, metabelian, soluble, monomial

Aliases: C₂²×C₃²⋊A₄, C₆²⋊₁A₄, C₂⁴⋊₂He₃, (C₆×A₄)⋊₂C₆, C₂³⋊(C₂×He₃), C₆.₁₇(C₆×A₄), (C₂×C₆²)⋊₇C₆, C₆²⋊₁₄(C₂×C₆), (C₂×C₆).₆C₆², C₂²⋊(C₂²×He₃), (C₂²×C₆²)⋊₁C₃, C₃²⋊₃(C₂²×A₄), (C₂³×C₆).₁₀C₃², (A₄×C₂×C₆)⋊₂C₃, C₃.₅(A₄×C₂×C₆), (C₃×C₆)⋊₂(C₂×A₄), (C₃×A₄)⋊₃(C₂×C₆), (C₂×C₆).₂₆(C₃×A₄), (C₂²×C₆).₁₁(C₃×C₆), SmallGroup(432,550)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₆ — C₂²×C₃²⋊A₄

Chief series

C₁ — C₂² — C₂×C₆ — C₆² — C₃²⋊A₄ — C₂×C₃²⋊A₄ — C₂²×C₃²⋊A₄

Lower central

C₂² — C₂×C₆ — C₂²×C₃²⋊A₄

Upper central

C₁ — C₂×C₆ — C₆²

Generators and relations for C₂²×C₃²⋊A₄
G = < a,b,c,d,e,f,g | a²=b²=c³=d³=e²=f²=g³=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, bd=db, be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, gcg^-1=cd^-1, de=ed, df=fd, dg=gd, geg^-1=ef=fe, gfg^-1=e >

Subgroups: 727 in 224 conjugacy classes, 50 normal (14 characteristic)
C₁, C₂, C₂, C₃, C₃, C₂², C₂², C₆, C₆, C₂³, C₂³, C₃², C₃², A₄, C₂×C₆, C₂×C₆, C₂⁴, C₃×C₆, C₃×C₆, C₂×A₄, C₂²×C₆, C₂²×C₆, He₃, C₃×A₄, C₆², C₆², C₂²×A₄, C₂³×C₆, C₂³×C₆, C₂×He₃, C₆×A₄, C₂×C₆², C₂×C₆², C₃²⋊A₄, C₂²×He₃, A₄×C₂×C₆, C₂²×C₆², C₂×C₃²⋊A₄, C₂²×C₃²⋊A₄
Quotients: C₁, C₂, C₃, C₂², C₆, C₃², A₄, C₂×C₆, C₃×C₆, C₂×A₄, He₃, C₃×A₄, C₆², C₂²×A₄, C₂×He₃, C₆×A₄, C₃²⋊A₄, C₂²×He₃, A₄×C₂×C₆, C₂×C₃²⋊A₄, C₂²×C₃²⋊A₄

Smallest permutation representation of C₂²×C₃²⋊A₄
►On 36 points

Generators in S₃₆

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

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

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

(1 2)(3 4)(5 6)(7 12)(8 9)(10 11)(16 23)(17 24)(18 22)(28 32)(29 33)(30 31)

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

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

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

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

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

80 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	3A	3B	3C	3D	3E	···	3J	6A	···	6F	6G	···	6AR	6AS	···	6BJ
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	3	3	3	3	1	1	3	3	12	···	12	1	···	1	3	···	3	12	···	12

80 irreducible representations

dim	1	1	1	1	1	1	3	3	3	3	3	3	3	3
type	+	+					+	+
image	C₁	C₂	C₃	C₃	C₆	C₆	A₄	C₂×A₄	He₃	C₃×A₄	C₂×He₃	C₆×A₄	C₃²⋊A₄	C₂×C₃²⋊A₄
kernel	C₂²×C₃²⋊A₄	C₂×C₃²⋊A₄	A₄×C₂×C₆	C₂²×C₆²	C₆×A₄	C₂×C₆²	C₆²	C₃×C₆	C₂⁴	C₂×C₆	C₂³	C₆	C₂²	C₂
# reps	1	3	6	2	18	6	1	3	2	2	6	6	6	18

Matrix representation of C₂²×C₃²⋊A₄ ►in GL₄(𝔽₇) generated by

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

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

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

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

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

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

1	0	0	0
0	0	1	0
0	0	0	1
0	1	0	0

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

C₂²×C₃²⋊A₄ in GAP, Magma, Sage, TeX

C_2^2\times C_3^2\rtimes A_4

% in TeX

G:=Group("C2^2xC3^2:A4");

// GroupNames label

G:=SmallGroup(432,550);

// by ID

G=gap.SmallGroup(432,550);

# by ID

G:=PCGroup([7,-2,-2,-3,-3,-3,-2,2,353,2287,3989]);

// Polycyclic

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

// generators/relations

G = C22×C32⋊A4 order 432 = 24·33

Direct product of C22 and C32⋊A4

G = C₂²×C₃²⋊A₄ order 432 = 2⁴·3³

Direct product of C₂² and C₃²⋊A₄