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G = C4×C32.A4order 432 = 24·33

Direct product of C4 and C32.A4

direct product, metabelian, soluble, monomial

Aliases: C4×C32.A4, C62.12C12, C6.9(C6×A4), C32.(C4×A4), C3.A43C12, C3.4(C12×A4), C12.6(C3×A4), (C3×C12).1A4, (C2×C62).12C6, (C22×C12).4C32, (C22×C4)⋊23- 1+2, C222(C4×3- 1+2), C23.2(C2×3- 1+2), (C2×C6×C12).2C3, (C4×C3.A4)⋊2C3, (C3×C6).4(C2×A4), (C2×C6).4(C3×C12), (C2×C3.A4).3C6, (C22×C6).5(C3×C6), C2.1(C2×C32.A4), (C2×C32.A4).2C2, SmallGroup(432,332)

Series: Derived Chief Lower central Upper central

C1C2×C6 — C4×C32.A4
C1C22C2×C6C22×C6C2×C62C2×C32.A4 — C4×C32.A4
C22C2×C6 — C4×C32.A4
C1C12C3×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 [×2], C3, C3, C4, C4, C22, C22 [×2], C6, C6 [×9], C2×C4 [×2], C23, C9 [×3], C32, C12, C12 [×5], C2×C6, C2×C6 [×9], C22×C4, C18 [×3], C3×C6, C3×C6 [×2], C2×C12 [×8], C22×C6, C22×C6, 3- 1+2, C36 [×3], C3.A4 [×3], C3×C12, C3×C12, C62, C62 [×2], C22×C12, C22×C12, C2×3- 1+2, C2×C3.A4 [×3], C6×C12 [×2], C2×C62, C4×3- 1+2, C32.A4, C4×C3.A4 [×3], C2×C6×C12, C2×C32.A4, C4×C32.A4
Quotients: C1, C2, C3 [×4], C4, C6 [×4], C32, C12 [×4], 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 23 31)(2 17 24 32)(3 18 25 33)(4 10 26 34)(5 11 27 35)(6 12 19 36)(7 13 20 28)(8 14 21 29)(9 15 22 30)
(2 8 5)(3 6 9)(11 17 14)(12 15 18)(19 22 25)(21 27 24)(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 24)(3 25)(5 27)(6 19)(8 21)(9 22)(11 35)(12 36)(14 29)(15 30)(17 32)(18 33)
(1 23)(3 25)(4 26)(6 19)(7 20)(9 22)(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)| (1,16,23,31)(2,17,24,32)(3,18,25,33)(4,10,26,34)(5,11,27,35)(6,12,19,36)(7,13,20,28)(8,14,21,29)(9,15,22,30), (2,8,5)(3,6,9)(11,17,14)(12,15,18)(19,22,25)(21,27,24)(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), (2,24)(3,25)(5,27)(6,19)(8,21)(9,22)(11,35)(12,36)(14,29)(15,30)(17,32)(18,33), (1,23)(3,25)(4,26)(6,19)(7,20)(9,22)(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( (1,16,23,31)(2,17,24,32)(3,18,25,33)(4,10,26,34)(5,11,27,35)(6,12,19,36)(7,13,20,28)(8,14,21,29)(9,15,22,30), (2,8,5)(3,6,9)(11,17,14)(12,15,18)(19,22,25)(21,27,24)(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), (2,24)(3,25)(5,27)(6,19)(8,21)(9,22)(11,35)(12,36)(14,29)(15,30)(17,32)(18,33), (1,23)(3,25)(4,26)(6,19)(7,20)(9,22)(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([(1,16,23,31),(2,17,24,32),(3,18,25,33),(4,10,26,34),(5,11,27,35),(6,12,19,36),(7,13,20,28),(8,14,21,29),(9,15,22,30)], [(2,8,5),(3,6,9),(11,17,14),(12,15,18),(19,22,25),(21,27,24),(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)], [(2,24),(3,25),(5,27),(6,19),(8,21),(9,22),(11,35),(12,36),(14,29),(15,30),(17,32),(18,33)], [(1,23),(3,25),(4,26),(6,19),(7,20),(9,22),(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 2A2B2C3A3B3C3D4A4B4C4D6A6B6C···6T9A···9F12A12B12C12D12E···12X18A···18F36A···36L
order122233334444666···69···91212121212···1218···1836···36
size113311331133113···312···1211113···312···1212···12

80 irreducible representations

dim111111111333333333333
type++++
imageC1C2C3C3C4C6C6C12C12A4C2×A43- 1+2C3×A4C4×A4C2×3- 1+2C6×A4C32.A4C4×3- 1+2C12×A4C2×C32.A4C4×C32.A4
kernelC4×C32.A4C2×C32.A4C4×C3.A4C2×C6×C12C32.A4C2×C3.A4C2×C62C3.A4C62C3×C12C3×C6C22×C4C12C32C23C6C4C22C3C2C1
# reps11622621241122222644612

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

500
050
005
,
900
030
001
,
300
030
003
,
1200
0120
001
,
100
0120
0012
,
007
300
020
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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