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## G = C3×S3×A4order 216 = 23·33

### Direct product of C3, S3 and A4

Aliases: C3×S3×A4, C625C6, C3⋊(C6×A4), (C3×A4)⋊5C6, C323(C2×A4), (C32×A4)⋊4C2, (C22×S3)⋊C32, C222(S3×C32), (S3×C2×C6)⋊C3, (C2×C6)⋊(C3×C6), (C2×C6)⋊4(C3×S3), SmallGroup(216,166)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C3×S3×A4
 Chief series C1 — C3 — C2×C6 — C62 — C32×A4 — C3×S3×A4
 Lower central C2×C6 — C3×S3×A4
 Upper central C1 — C3

Generators and relations for C3×S3×A4
G = < a,b,c,d,e,f | a3=b3=c2=d2=e2=f3=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, fdf-1=de=ed, fef-1=d >

Subgroups: 308 in 81 conjugacy classes, 24 normal (15 characteristic)
C1, C2, C3, C3, C22, C22, S3, S3, C6, C23, C32, C32, A4, A4, D6, C2×C6, C2×C6, C3×S3, C3×S3, C3×C6, C2×A4, C22×S3, C22×C6, C33, C3×A4, C3×A4, C3×A4, S3×C6, C62, S3×C32, S3×A4, C6×A4, S3×C2×C6, C32×A4, C3×S3×A4
Quotients: C1, C2, C3, S3, C6, C32, A4, C3×S3, C3×C6, C2×A4, C3×A4, S3×C32, S3×A4, C6×A4, C3×S3×A4

Permutation representations of C3×S3×A4
On 24 points - transitive group 24T563
Generators in S24
(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 2 3)(4 5 6)(7 8 9)(10 11 12)(13 15 14)(16 18 17)(19 21 20)(22 24 23)
(1 13)(2 14)(3 15)(4 16)(5 17)(6 18)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)
(1 10)(2 11)(3 12)(4 7)(5 8)(6 9)(13 22)(14 23)(15 24)(16 19)(17 20)(18 21)
(1 4)(2 5)(3 6)(7 10)(8 11)(9 12)(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)
(1 2 3)(4 8 12)(5 9 10)(6 7 11)(13 14 15)(16 20 24)(17 21 22)(18 19 23)

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

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

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

G:=TransitiveGroup(24,563);

36 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 3F ··· 3K 3L ··· 3Q 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J ··· 6O order 1 2 2 2 3 3 3 3 3 3 ··· 3 3 ··· 3 6 6 6 6 6 6 6 6 6 6 ··· 6 size 1 3 3 9 1 1 2 2 2 4 ··· 4 8 ··· 8 3 3 3 3 6 6 6 9 9 12 ··· 12

36 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 3 3 3 3 6 6 type + + + + + + image C1 C2 C3 C3 C6 C6 S3 C3×S3 C3×S3 A4 C2×A4 C3×A4 C6×A4 S3×A4 C3×S3×A4 kernel C3×S3×A4 C32×A4 S3×A4 S3×C2×C6 C3×A4 C62 C3×A4 A4 C2×C6 C3×S3 C32 S3 C3 C3 C1 # reps 1 1 6 2 6 2 1 6 2 1 1 2 2 1 2

Matrix representation of C3×S3×A4 in GL5(𝔽7)

 4 0 0 0 0 0 4 0 0 0 0 0 4 0 0 0 0 0 4 0 0 0 0 0 4
,
 2 0 0 0 0 0 4 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 0 6 0 0 0 6 0 0 0 0 0 0 6 0 0 0 0 0 6 0 0 0 0 0 6
,
 1 0 0 0 0 0 1 0 0 0 0 0 6 6 6 0 0 0 0 1 0 0 0 1 0
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 6 6 6
,
 4 0 0 0 0 0 4 0 0 0 0 0 2 0 0 0 0 5 5 5 0 0 0 2 0

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

C3×S3×A4 in GAP, Magma, Sage, TeX

C_3\times S_3\times A_4
% in TeX

G:=Group("C3xS3xA4");
// GroupNames label

G:=SmallGroup(216,166);
// by ID

G=gap.SmallGroup(216,166);
# by ID

G:=PCGroup([6,-2,-3,-3,-2,2,-3,657,280,5189]);
// Polycyclic

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

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