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## G = C4×C32.A4order 432 = 24·33

### Direct product of C4 and C32.A4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C4×C32.A4
 Chief series C1 — C22 — C2×C6 — C22×C6 — C2×C62 — C2×C32.A4 — C4×C32.A4
 Lower central C22 — C2×C6 — C4×C32.A4
 Upper central C1 — C12 — C3×C12

Generators and relations for C4×C32.A4
G = < a,b,c,d,e,f | a4=b3=c3=d2=e2=1, f3=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf-1=bc-1, cd=dc, ce=ec, cf=fc, fdf-1=de=ed, fef-1=d >

Subgroups: 222 in 90 conjugacy classes, 30 normal (21 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, C6, C6, C2×C4, C23, C9, C32, C12, C12, C2×C6, C2×C6, C22×C4, C18, C3×C6, C3×C6, C2×C12, C22×C6, C22×C6, 3- 1+2, C36, C3.A4, C3×C12, C3×C12, C62, C62, C22×C12, C22×C12, C2×3- 1+2, C2×C3.A4, C6×C12, C2×C62, C4×3- 1+2, C32.A4, C4×C3.A4, C2×C6×C12, C2×C32.A4, C4×C32.A4
Quotients: C1, C2, C3, C4, C6, C32, C12, A4, C3×C6, C2×A4, 3- 1+2, C3×C12, C3×A4, C4×A4, C2×3- 1+2, C6×A4, C4×3- 1+2, C32.A4, C12×A4, C2×C32.A4, C4×C32.A4

Smallest permutation representation of C4×C32.A4
On 36 points
Generators in S36
(1 16 19 31)(2 17 20 32)(3 18 21 33)(4 10 22 34)(5 11 23 35)(6 12 24 36)(7 13 25 28)(8 14 26 29)(9 15 27 30)
(2 8 5)(3 6 9)(11 17 14)(12 15 18)(20 26 23)(21 24 27)(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)
(2 20)(3 21)(5 23)(6 24)(8 26)(9 27)(11 35)(12 36)(14 29)(15 30)(17 32)(18 33)
(1 19)(3 21)(4 22)(6 24)(7 25)(9 27)(10 34)(12 36)(13 28)(15 30)(16 31)(18 33)
(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)| (1,16,19,31)(2,17,20,32)(3,18,21,33)(4,10,22,34)(5,11,23,35)(6,12,24,36)(7,13,25,28)(8,14,26,29)(9,15,27,30), (2,8,5)(3,6,9)(11,17,14)(12,15,18)(20,26,23)(21,24,27)(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), (2,20)(3,21)(5,23)(6,24)(8,26)(9,27)(11,35)(12,36)(14,29)(15,30)(17,32)(18,33), (1,19)(3,21)(4,22)(6,24)(7,25)(9,27)(10,34)(12,36)(13,28)(15,30)(16,31)(18,33), (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( (1,16,19,31)(2,17,20,32)(3,18,21,33)(4,10,22,34)(5,11,23,35)(6,12,24,36)(7,13,25,28)(8,14,26,29)(9,15,27,30), (2,8,5)(3,6,9)(11,17,14)(12,15,18)(20,26,23)(21,24,27)(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), (2,20)(3,21)(5,23)(6,24)(8,26)(9,27)(11,35)(12,36)(14,29)(15,30)(17,32)(18,33), (1,19)(3,21)(4,22)(6,24)(7,25)(9,27)(10,34)(12,36)(13,28)(15,30)(16,31)(18,33), (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([[(1,16,19,31),(2,17,20,32),(3,18,21,33),(4,10,22,34),(5,11,23,35),(6,12,24,36),(7,13,25,28),(8,14,26,29),(9,15,27,30)], [(2,8,5),(3,6,9),(11,17,14),(12,15,18),(20,26,23),(21,24,27),(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)], [(2,20),(3,21),(5,23),(6,24),(8,26),(9,27),(11,35),(12,36),(14,29),(15,30),(17,32),(18,33)], [(1,19),(3,21),(4,22),(6,24),(7,25),(9,27),(10,34),(12,36),(13,28),(15,30),(16,31),(18,33)], [(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 3A 3B 3C 3D 4A 4B 4C 4D 6A 6B 6C ··· 6T 9A ··· 9F 12A 12B 12C 12D 12E ··· 12X 18A ··· 18F 36A ··· 36L order 1 2 2 2 3 3 3 3 4 4 4 4 6 6 6 ··· 6 9 ··· 9 12 12 12 12 12 ··· 12 18 ··· 18 36 ··· 36 size 1 1 3 3 1 1 3 3 1 1 3 3 1 1 3 ··· 3 12 ··· 12 1 1 1 1 3 ··· 3 12 ··· 12 12 ··· 12

80 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 3 3 3 3 3 3 3 3 3 3 3 3 type + + + + image C1 C2 C3 C3 C4 C6 C6 C12 C12 A4 C2×A4 3- 1+2 C3×A4 C4×A4 C2×3- 1+2 C6×A4 C32.A4 C4×3- 1+2 C12×A4 C2×C32.A4 C4×C32.A4 kernel C4×C32.A4 C2×C32.A4 C4×C3.A4 C2×C6×C12 C32.A4 C2×C3.A4 C2×C62 C3.A4 C62 C3×C12 C3×C6 C22×C4 C12 C32 C23 C6 C4 C22 C3 C2 C1 # reps 1 1 6 2 2 6 2 12 4 1 1 2 2 2 2 2 6 4 4 6 12

Matrix representation of C4×C32.A4 in GL3(𝔽13) generated by

 5 0 0 0 5 0 0 0 5
,
 9 0 0 0 3 0 0 0 1
,
 3 0 0 0 3 0 0 0 3
,
 12 0 0 0 12 0 0 0 1
,
 1 0 0 0 12 0 0 0 12
,
 0 0 7 3 0 0 0 2 0
G:=sub<GL(3,GF(13))| [5,0,0,0,5,0,0,0,5],[9,0,0,0,3,0,0,0,1],[3,0,0,0,3,0,0,0,3],[12,0,0,0,12,0,0,0,1],[1,0,0,0,12,0,0,0,12],[0,3,0,0,0,2,7,0,0] >;

C4×C32.A4 in GAP, Magma, Sage, TeX

C_4\times C_3^2.A_4
% in TeX

G:=Group("C4xC3^2.A4");
// GroupNames label

G:=SmallGroup(432,332);
// by ID

G=gap.SmallGroup(432,332);
# by ID

G:=PCGroup([7,-2,-3,-3,-2,-3,-2,2,126,260,450,4548,7951]);
// Polycyclic

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

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