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, C₆².₇A₄, C₂⁴⋊₅3_-¹⁺², C₆.₁₆(C₆×A₄), (C₂×C₆).₅C₆², C₃².(C₂²×A₄), (C₂×C₆²).₁₇C₆, C₆².₄₃(C₂×C₆), (C₂²×C₆²).₂C₃, (C₂³×C₆).₉C₃², C₂³⋊₂(C₂×3_-¹⁺²), C₂²⋊₂(C₂²×3_-¹⁺²), C₃.₄(A₄×C₂×C₆), (C₂×C₃.A₄)⋊₃C₆, C₃.A₄⋊₃(C₂×C₆), (C₂×C₆).₂₅(C₃×A₄), (C₃×C₆).₁₀(C₂×A₄), (C₂²×C₃.A₄)⋊₆C₃, (C₂²×C₆).₁₀(C₃×C₆), SmallGroup(432,549)

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=1, g³=d, 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: 502 in 194 conjugacy classes, 50 normal (14 characteristic)
C₁, C₂, C₂, C₃, C₃, C₂², C₂², C₆, C₆, C₂³, C₂³, C₉, C₃², C₂×C₆, C₂×C₆, C₂⁴, C₁₈, C₃×C₆, C₃×C₆, C₂²×C₆, C₂²×C₆, 3_-¹⁺², C₃.A₄, C₂×C₁₈, C₆², C₆², C₂³×C₆, C₂³×C₆, C₂×3_-¹⁺², C₂×C₃.A₄, C₂×C₆², C₂×C₆², C₃².A₄, C₂²×3_-¹⁺², C₂²×C₃.A₄, C₂²×C₆², C₂×C₃².A₄, C₂²×C₃².A₄
Quotients: C₁, C₂, C₃, C₂², C₆, C₃², A₄, C₂×C₆, C₃×C₆, C₂×A₄, 3_-¹⁺², C₃×A₄, C₆², C₂²×A₄, C₂×3_-¹⁺², C₆×A₄, C₃².A₄, C₂²×3_-¹⁺², A₄×C₂×C₆, C₂×C₃².A₄, C₂²×C₃².A₄

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

Generators in S₃₆

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

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

(2 8 5)(3 6 9)(11 17 14)(12 15 18)(19 25 22)(20 23 26)(29 35 32)(30 33 36)

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

(1 21)(3 23)(4 24)(6 26)(7 27)(9 20)(10 34)(12 36)(13 28)(15 30)(16 31)(18 33)

(1 21)(2 22)(4 24)(5 25)(7 27)(8 19)(10 34)(11 35)(13 28)(14 29)(16 31)(17 32)

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

80 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	3A	3B	3C	3D	6A	···	6F	6G	···	6AR	9A	···	9F	18A	···	18R
order	1	2	2	2	2	2	2	2	3	3	3	3	6	···	6	6	···	6	9	···	9	18	···	18
size	1	1	1	1	3	3	3	3	1	1	3	3	1	···	1	3	···	3	12	···	12	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	3_-¹⁺²	C₃×A₄	C₂×3_-¹⁺²	C₆×A₄	C₃².A₄	C₂×C₃².A₄
kernel	C₂²×C₃².A₄	C₂×C₃².A₄	C₂²×C₃.A₄	C₂²×C₆²	C₂×C₃.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 GL₆(𝔽₁₉)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	0
0	0	0	18	0	0
0	0	0	0	18	0
0	0	0	0	0	18

18	0	0	0	0	0
0	18	0	0	0	0
0	0	18	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	11	0	0	0	0
0	0	7	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1

11	0	0	0	0	0
0	11	0	0	0	0
0	0	11	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	18	0	0
0	0	0	0	18	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	18	0	0
0	0	0	0	1	0
0	0	0	0	0	18

0	1	0	0	0	0
0	0	1	0	0	0
11	0	0	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1
0	0	0	1	0	0

G:=sub<GL(6,GF(19))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,18],[18,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,11,0,0,0,0,0,0,7,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,18,0,0,0,0,0,0,1,0,0,0,0,0,0,18],[0,0,11,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0] >;

C₂²×C₃².A₄ in GAP, Magma, Sage, TeX

C_2^2\times C_3^2.A_4

% in TeX

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

// GroupNames label

G:=SmallGroup(432,549);

// by ID

G=gap.SmallGroup(432,549);

# by ID

G:=PCGroup([7,-2,-2,-3,-3,-3,-2,2,205,353,2287,3989]);

// Polycyclic

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