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## G = C3×S32⋊C4order 432 = 24·33

### Direct product of C3 and S32⋊C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3⋊S3 — C3×S32⋊C4
 Chief series C1 — C32 — C3×C6 — C2×C3⋊S3 — C6×C3⋊S3 — S32×C6 — C3×S32⋊C4
 Lower central C32 — C3⋊S3 — C3×S32⋊C4
 Upper central C1 — C6

Generators and relations for C3×S32⋊C4
G = < a,b,c,d,e,f | a3=b3=c2=d3=e2=f4=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, cbc=fdf-1=b-1, bd=db, be=eb, fbf-1=ede=d-1, cd=dc, ce=ec, fcf-1=e, fef-1=c >

Subgroups: 652 in 132 conjugacy classes, 26 normal (22 characteristic)
C1, C2, C2 [×4], C3, C3 [×4], C4 [×2], C22 [×5], S3 [×6], C6, C6 [×12], C2×C4 [×2], C23, C32, C32 [×4], Dic3, C12 [×4], D6 [×7], C2×C6 [×7], C22⋊C4, C3×S3 [×10], C3⋊S3 [×2], C3×C6, C3×C6 [×6], C4×S3, C2×C12 [×2], C22×S3, C22×C6, C33, C3×Dic3 [×3], C3×C12, C32⋊C4, S32 [×2], S32, S3×C6 [×9], C2×C3⋊S3, C62, C3×C22⋊C4, S3×C32 [×2], C3×C3⋊S3 [×2], C32×C6, C6.D6, S3×C12, C2×C32⋊C4, C2×S32, S3×C2×C6, C32×Dic3, C3×C32⋊C4, C3×S32 [×2], C3×S32, S3×C3×C6, C6×C3⋊S3, S32⋊C4, C3×C6.D6, C6×C32⋊C4, S32×C6, C3×S32⋊C4
Quotients: C1, C2 [×3], C3, C4 [×2], C22, C6 [×3], C2×C4, D4 [×2], C12 [×2], C2×C6, C22⋊C4, C2×C12, C3×D4 [×2], C3×C22⋊C4, S3≀C2, S32⋊C4, C3×S3≀C2, C3×S32⋊C4

Permutation representations of C3×S32⋊C4
On 24 points - transitive group 24T1317
Generators in S24
(1 9 5)(2 10 6)(3 11 7)(4 12 8)(13 19 23)(14 20 24)(15 17 21)(16 18 22)
(2 6 10)(4 8 12)(14 20 24)(16 18 22)
(2 18)(4 20)(6 16)(8 14)(10 22)(12 24)
(1 9 5)(3 11 7)(13 23 19)(15 21 17)
(1 17)(3 19)(5 15)(7 13)(9 21)(11 23)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)

G:=sub<Sym(24)| (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,19,23)(14,20,24)(15,17,21)(16,18,22), (2,6,10)(4,8,12)(14,20,24)(16,18,22), (2,18)(4,20)(6,16)(8,14)(10,22)(12,24), (1,9,5)(3,11,7)(13,23,19)(15,21,17), (1,17)(3,19)(5,15)(7,13)(9,21)(11,23), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)>;

G:=Group( (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,19,23)(14,20,24)(15,17,21)(16,18,22), (2,6,10)(4,8,12)(14,20,24)(16,18,22), (2,18)(4,20)(6,16)(8,14)(10,22)(12,24), (1,9,5)(3,11,7)(13,23,19)(15,21,17), (1,17)(3,19)(5,15)(7,13)(9,21)(11,23), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24) );

G=PermutationGroup([(1,9,5),(2,10,6),(3,11,7),(4,12,8),(13,19,23),(14,20,24),(15,17,21),(16,18,22)], [(2,6,10),(4,8,12),(14,20,24),(16,18,22)], [(2,18),(4,20),(6,16),(8,14),(10,22),(12,24)], [(1,9,5),(3,11,7),(13,23,19),(15,21,17)], [(1,17),(3,19),(5,15),(7,13),(9,21),(11,23)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)])

G:=TransitiveGroup(24,1317);

54 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C ··· 3H 4A 4B 4C 4D 6A 6B 6C ··· 6H 6I 6J 6K 6L 6M 6N 6O 6P 6Q ··· 6V 12A 12B 12C 12D 12E ··· 12J 12K 12L 12M 12N order 1 2 2 2 2 2 3 3 3 ··· 3 4 4 4 4 6 6 6 ··· 6 6 6 6 6 6 6 6 6 6 ··· 6 12 12 12 12 12 ··· 12 12 12 12 12 size 1 1 6 6 9 9 1 1 4 ··· 4 6 6 18 18 1 1 4 ··· 4 6 6 6 6 9 9 9 9 12 ··· 12 6 6 6 6 12 ··· 12 18 18 18 18

54 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 4 4 4 4 4 type + + + + + + + + image C1 C2 C2 C2 C3 C4 C6 C6 C6 C12 D4 D4 C3×D4 C3×D4 S3≀C2 S32⋊C4 S32⋊C4 C3×S3≀C2 C3×S32⋊C4 kernel C3×S32⋊C4 C3×C6.D6 C6×C32⋊C4 S32×C6 S32⋊C4 C3×S32 C6.D6 C2×C32⋊C4 C2×S32 S32 C3×C3⋊S3 C32×C6 C3⋊S3 C3×C6 C6 C3 C3 C2 C1 # reps 1 1 1 1 2 4 2 2 2 8 1 1 2 2 4 2 2 8 8

Matrix representation of C3×S32⋊C4 in GL4(𝔽7) generated by

 4 0 0 0 0 4 0 0 0 0 4 0 0 0 0 4
,
 3 5 5 1 3 1 1 5 6 4 6 3 1 6 6 5
,
 6 4 0 6 5 5 0 6 2 3 1 1 6 2 0 4
,
 4 2 5 0 2 3 5 2 2 2 6 2 2 4 3 2
,
 5 1 6 2 6 6 2 3 6 5 3 3 2 4 3 2
,
 3 4 4 3 4 1 3 6 4 0 4 3 0 4 3 6
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[3,3,6,1,5,1,4,6,5,1,6,6,1,5,3,5],[6,5,2,6,4,5,3,2,0,0,1,0,6,6,1,4],[4,2,2,2,2,3,2,4,5,5,6,3,0,2,2,2],[5,6,6,2,1,6,5,4,6,2,3,3,2,3,3,2],[3,4,4,0,4,1,0,4,4,3,4,3,3,6,3,6] >;

C3×S32⋊C4 in GAP, Magma, Sage, TeX

C_3\times S_3^2\rtimes C_4
% in TeX

G:=Group("C3xS3^2:C4");
// GroupNames label

G:=SmallGroup(432,574);
// by ID

G=gap.SmallGroup(432,574);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,3,197,176,4037,3036,362,1189,1203]);
// Polycyclic

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

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