Copied to
clipboard

## G = C22×C32.A4order 432 = 24·33

### Direct product of C22 and C32.A4

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

׿
×
𝔽