C3xA4^2 - GroupNames

G = C₃xA₄² order 432 = 2⁴·3³

Direct product of C₃, A₄ and A₄

direct product, metabelian, soluble, monomial, A-group

Aliases: C₃xA₄², C₂⁴:C₃³, C₂²:A₄:C₃², (C₂³xC₆):C₃², (C₂²xA₄):C₃², C₂²:₁(C₃²xA₄), (A₄xC₂xC₆):₃C₃, (C₂xC₆):₁(C₃xA₄), (C₃xC₂²:A₄):₃C₃, SmallGroup(432,750)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂⁴ — C₃xA₄²

Chief series

C₁ — C₂² — C₂⁴ — C₂²xA₄ — A₄² — C₃xA₄²

Lower central

C₂⁴ — C₃xA₄²

Upper central

C₁ — C₃

Generators and relations for C₃xA₄²
G = < a,b,c,d,e,f,g | a³=b²=c²=d³=e²=f²=g³=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, dbd^-1=bc=cb, be=eb, bf=fb, bg=gb, dcd^-1=b, ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, geg^-1=ef=fe, gfg^-1=e >

Subgroups: 880 in 158 conjugacy classes, 42 normal (5 characteristic)
C₁, C₂, C₃, C₃, C₂², C₂², C₆, C₂³, C₃², A₄, A₄, C₂xC₆, C₂xC₆, C₂⁴, C₃xC₆, C₂xA₄, C₂²xC₆, C₃³, C₃xA₄, C₃xA₄, C₆², C₂²xA₄, C₂²:A₄, C₂³xC₆, C₆xA₄, C₃²xA₄, A₄², A₄xC₂xC₆, C₃xC₂²:A₄, C₃xA₄²
Quotients: C₁, C₃, C₃², A₄, C₃³, C₃xA₄, C₃²xA₄, A₄², C₃xA₄²

Smallest permutation representation of C₃xA₄²
►On 36 points

Generators in S₃₆

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

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

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

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

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

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

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

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

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

48 conjugacy classes

class	1	2A	2B	2C	3A	3B	3C	···	3N	3O	···	3Z	6A	6B	6C	6D	6E	6F	6G	···	6R
order	1	2	2	2	3	3	3	···	3	3	···	3	6	6	6	6	6	6	6	···	6
size	1	3	3	9	1	1	4	···	4	16	···	16	3	3	3	3	9	9	12	···	12

48 irreducible representations

dim	1	1	1	1	3	3	3	9	9
type	+				+			+
image	C₁	C₃	C₃	C₃	A₄	C₃xA₄	C₃xA₄	A₄²	C₃xA₄²
kernel	C₃xA₄²	A₄²	A₄xC₂xC₆	C₃xC₂²:A₄	C₃xA₄	A₄	C₂xC₆	C₃	C₁
# reps	1	18	4	4	2	12	4	1	2

Matrix representation of C₃xA₄² ►in GL₆(F₇)

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

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

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

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

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

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

4	0	0	0	0	0
0	0	4	0	0	0
3	3	3	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1

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

C₃xA₄² in GAP, Magma, Sage, TeX

C_3\times A_4^2

% in TeX

G:=Group("C3xA4^2");

// GroupNames label

G:=SmallGroup(432,750);

// by ID

G=gap.SmallGroup(432,750);

# by ID

G:=PCGroup([7,-3,-3,-3,-2,2,-2,2,766,326,13613,5298]);

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^3=b^2=c^2=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,d*b*d^-1=b*c=c*b,b*e=e*b,b*f=f*b,b*g=g*b,d*c*d^-1=b,c*e=e*c,c*f=f*c,c*g=g*c,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 = C3xA42 order 432 = 24·33

Direct product of C3, A4 and A4

G = C₃xA₄² order 432 = 2⁴·3³

Direct product of C₃, A₄ and A₄