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## G = C3×S3×C3⋊D4order 432 = 24·33

### Direct product of C3, S3 and C3⋊D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C3×S3×C3⋊D4
 Chief series C1 — C3 — C32 — C3×C6 — C32×C6 — S3×C3×C6 — S32×C6 — C3×S3×C3⋊D4
 Lower central C32 — C3×C6 — C3×S3×C3⋊D4
 Upper central C1 — C6 — C2×C6

Generators and relations for C3×S3×C3⋊D4
G = < a,b,c,d,e,f | a3=b3=c2=d3=e4=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, cbc=b-1, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, ede-1=fdf=d-1, fef=e-1 >

Subgroups: 1032 in 290 conjugacy classes, 72 normal (64 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, S3, C6, C6, C2×C4, D4, C23, C32, C32, Dic3, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×D4, C3×S3, C3×S3, C3⋊S3, C3×C6, C3×C6, C4×S3, D12, C2×Dic3, C3⋊D4, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×S3, C22×C6, C33, C3×Dic3, C3×Dic3, C3⋊Dic3, C3×C12, S32, S3×C6, S3×C6, C2×C3⋊S3, C62, C62, S3×D4, C2×C3⋊D4, C6×D4, S3×C32, S3×C32, C3×C3⋊S3, C32×C6, C32×C6, S3×Dic3, D6⋊S3, C3⋊D12, S3×C12, C3×D12, C6×Dic3, C3×C3⋊D4, C3×C3⋊D4, C327D4, D4×C32, C2×S32, S3×C2×C6, S3×C2×C6, C2×C62, C32×Dic3, C3×C3⋊Dic3, C3×S32, S3×C3×C6, S3×C3×C6, C6×C3⋊S3, C3×C62, S3×C3⋊D4, C3×S3×D4, C6×C3⋊D4, C3×S3×Dic3, C3×D6⋊S3, C3×C3⋊D12, C32×C3⋊D4, C3×C327D4, S32×C6, S3×C62, C3×S3×C3⋊D4
Quotients: C1, C2, C3, C22, S3, C6, D4, C23, D6, C2×C6, C2×D4, C3×S3, C3⋊D4, C3×D4, C22×S3, C22×C6, S32, S3×C6, S3×D4, C2×C3⋊D4, C6×D4, C3×C3⋊D4, C2×S32, S3×C2×C6, C3×S32, S3×C3⋊D4, C3×S3×D4, C6×C3⋊D4, S32×C6, C3×S3×C3⋊D4

Permutation representations of C3×S3×C3⋊D4
On 24 points - transitive group 24T1280
Generators in S24
(1 16 17)(2 13 18)(3 14 19)(4 15 20)(5 10 23)(6 11 24)(7 12 21)(8 9 22)
(1 16 17)(2 13 18)(3 14 19)(4 15 20)(5 23 10)(6 24 11)(7 21 12)(8 22 9)
(1 22)(2 23)(3 24)(4 21)(5 13)(6 14)(7 15)(8 16)(9 17)(10 18)(11 19)(12 20)
(1 17 16)(2 13 18)(3 19 14)(4 15 20)(5 10 23)(6 24 11)(7 12 21)(8 22 9)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 4)(2 3)(5 6)(7 8)(9 12)(10 11)(13 14)(15 16)(17 20)(18 19)(21 22)(23 24)

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

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

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

G:=TransitiveGroup(24,1280);

81 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C ··· 3H 3I 3J 3K 4A 4B 6A 6B 6C ··· 6P 6Q 6R 6S 6T 6U ··· 6AF 6AG ··· 6AV 6AW 6AX 6AY 6AZ 6BA 12A 12B 12C 12D 12E 12F 12G order 1 2 2 2 2 2 2 2 3 3 3 ··· 3 3 3 3 4 4 6 6 6 ··· 6 6 6 6 6 6 ··· 6 6 ··· 6 6 6 6 6 6 12 12 12 12 12 12 12 size 1 1 2 3 3 6 6 18 1 1 2 ··· 2 4 4 4 6 18 1 1 2 ··· 2 3 3 3 3 4 ··· 4 6 ··· 6 12 12 12 18 18 6 6 12 12 12 18 18

81 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 C6 C6 S3 S3 D4 D6 D6 D6 C3×S3 C3×S3 C3⋊D4 C3×D4 S3×C6 S3×C6 S3×C6 C3×C3⋊D4 S32 S3×D4 C2×S32 C3×S32 S3×C3⋊D4 C3×S3×D4 S32×C6 C3×S3×C3⋊D4 kernel C3×S3×C3⋊D4 C3×S3×Dic3 C3×D6⋊S3 C3×C3⋊D12 C32×C3⋊D4 C3×C32⋊7D4 S32×C6 S3×C62 S3×C3⋊D4 S3×Dic3 D6⋊S3 C3⋊D12 C3×C3⋊D4 C32⋊7D4 C2×S32 S3×C2×C6 C3×C3⋊D4 S3×C2×C6 S3×C32 C3×Dic3 S3×C6 C62 C3⋊D4 C22×S3 C3×S3 C3×S3 Dic3 D6 C2×C6 S3 C2×C6 C32 C6 C22 C3 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 1 1 2 1 3 2 2 2 4 4 2 6 4 8 1 1 1 2 2 2 2 4

Matrix representation of C3×S3×C3⋊D4 in GL4(𝔽7) generated by

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

C3×S3×C3⋊D4 in GAP, Magma, Sage, TeX

C_3\times S_3\times C_3\rtimes D_4
% in TeX

G:=Group("C3xS3xC3:D4");
// GroupNames label

G:=SmallGroup(432,658);
// by ID

G=gap.SmallGroup(432,658);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,303,2028,14118]);
// Polycyclic

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

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