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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=f3=1, 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: 357 in 108 conjugacy classes, 30 normal (21 characteristic)
C1, C2, C2 [×2], C3, C3 [×4], C4, C4, C22, C22 [×2], C6, C6 [×12], C2×C4 [×2], C23, C32, C32 [×3], C12, C12 [×8], A4 [×3], C2×C6, C2×C6 [×9], C22×C4, C3×C6, C3×C6 [×5], C2×C12 [×8], C2×A4 [×3], C22×C6, C22×C6, He3, C3×C12, C3×C12 [×4], C3×A4 [×3], C62, C62 [×2], C4×A4 [×3], C22×C12, C22×C12, C2×He3, C6×C12 [×2], C6×A4 [×3], C2×C62, C4×He3, C32⋊A4, C12×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, He3, C3×C12, C3×A4, C4×A4, C2×He3, C6×A4, C4×He3, C32⋊A4, C12×A4, C2×C32⋊A4, C4×C32⋊A4

Smallest permutation representation of C4×C32⋊A4
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)
(5 26 32)(6 27 29)(7 28 30)(8 25 31)(9 20 36)(10 17 33)(11 18 34)(12 19 35)
(1 13 21)(2 14 22)(3 15 23)(4 16 24)(5 26 32)(6 27 29)(7 28 30)(8 25 31)(9 36 20)(10 33 17)(11 34 18)(12 35 19)
(5 7)(6 8)(9 11)(10 12)(17 19)(18 20)(25 27)(26 28)(29 31)(30 32)(33 35)(34 36)
(1 3)(2 4)(5 7)(6 8)(13 15)(14 16)(21 23)(22 24)(25 27)(26 28)(29 31)(30 32)
(1 35 8)(2 36 5)(3 33 6)(4 34 7)(9 32 22)(10 29 23)(11 30 24)(12 31 21)(13 19 25)(14 20 26)(15 17 27)(16 18 28)

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

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

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

80 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E ··· 3J 4A 4B 4C 4D 6A 6B 6C ··· 6T 6U ··· 6Z 12A 12B 12C 12D 12E ··· 12X 12Y ··· 12AJ order 1 2 2 2 3 3 3 3 3 ··· 3 4 4 4 4 6 6 6 ··· 6 6 ··· 6 12 12 12 12 12 ··· 12 12 ··· 12 size 1 1 3 3 1 1 3 3 12 ··· 12 1 1 3 3 1 1 3 ··· 3 12 ··· 12 1 1 1 1 3 ··· 3 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 He3 C3×A4 C4×A4 C2×He3 C6×A4 C32⋊A4 C4×He3 C12×A4 C2×C32⋊A4 C4×C32⋊A4 kernel C4×C32⋊A4 C2×C32⋊A4 C12×A4 C2×C6×C12 C32⋊A4 C6×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
,
 1 0 0 0 12 0 0 0 12
,
 12 0 0 0 1 0 0 0 12
,
 0 0 12 8 0 0 0 8 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],[1,0,0,0,12,0,0,0,12],[12,0,0,0,1,0,0,0,12],[0,8,0,0,0,8,12,0,0] >;

C4×C32⋊A4 in GAP, Magma, Sage, TeX

C_4\times C_3^2\rtimes A_4
% in TeX

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

G:=SmallGroup(432,333);
// by ID

G=gap.SmallGroup(432,333);
# by ID

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

G:=Group<a,b,c,d,e,f|a^4=b^3=c^3=d^2=e^2=f^3=1,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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