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## G = S3×C6.D6order 432 = 24·33

### Direct product of S3 and C6.D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C33 — S3×C6.D6
 Chief series C1 — C3 — C32 — C33 — C32×C6 — S3×C3×C6 — C3×S3×Dic3 — S3×C6.D6
 Lower central C33 — S3×C6.D6
 Upper central C1 — C2

Generators and relations for S3×C6.D6
G = < a,b,c,d,e | a3=b2=c6=e2=1, d6=c3, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=ece=c-1, ede=d5 >

Subgroups: 1788 in 290 conjugacy classes, 64 normal (18 characteristic)
C1, C2, C2 [×6], C3, C3 [×2], C3 [×4], C4 [×4], C22 [×7], S3 [×2], S3 [×20], C6, C6 [×2], C6 [×12], C2×C4 [×6], C23, C32, C32 [×2], C32 [×4], Dic3 [×2], Dic3 [×6], C12 [×8], D6, D6 [×26], C2×C6 [×4], C22×C4, C3×S3 [×4], C3×S3 [×8], C3⋊S3 [×2], C3⋊S3 [×14], C3×C6, C3×C6 [×2], C3×C6 [×6], C4×S3 [×14], C2×Dic3 [×3], C2×C12 [×3], C22×S3 [×4], C33, C3×Dic3 [×4], C3×Dic3 [×8], C3⋊Dic3 [×2], C3×C12 [×2], S32 [×12], S3×C6 [×2], S3×C6 [×4], C2×C3⋊S3, C2×C3⋊S3 [×11], C62, S3×C2×C4 [×3], S3×C32 [×2], C3×C3⋊S3 [×2], C33⋊C2 [×2], C32×C6, S3×Dic3 [×2], S3×Dic3 [×2], C6.D6, C6.D6 [×7], S3×C12 [×4], C6×Dic3 [×2], C4×C3⋊S3 [×2], C2×S32 [×3], C22×C3⋊S3, C32×Dic3 [×2], C3×C3⋊Dic3 [×2], S3×C3⋊S3 [×4], S3×C3×C6, C6×C3⋊S3, C2×C33⋊C2, C4×S32 [×2], C2×C6.D6, C3×S3×Dic3 [×2], C3×C6.D6, C338(C2×C4) [×2], C339(C2×C4), C2×S3×C3⋊S3, S3×C6.D6
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3 [×3], C2×C4 [×6], C23, D6 [×9], C22×C4, C4×S3 [×6], C22×S3 [×3], S32 [×3], S3×C2×C4 [×3], C6.D6 [×2], C2×S32 [×3], C4×S32 [×2], C2×C6.D6, S33, S3×C6.D6

Permutation representations of S3×C6.D6
On 24 points - transitive group 24T1298
Generators in S24
(1 5 9)(2 6 10)(3 7 11)(4 8 12)(13 21 17)(14 22 18)(15 23 19)(16 24 20)
(1 20)(2 21)(3 22)(4 23)(5 24)(6 13)(7 14)(8 15)(9 16)(10 17)(11 18)(12 19)
(1 11 9 7 5 3)(2 4 6 8 10 12)(13 15 17 19 21 23)(14 24 22 20 18 16)
(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 16)(2 21)(3 14)(4 19)(5 24)(6 17)(7 22)(8 15)(9 20)(10 13)(11 18)(12 23)

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

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

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

G:=TransitiveGroup(24,1298);

54 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 3E 3F 3G 4A 4B 4C 4D 4E 4F 4G 4H 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 6K 6L 6M 6N 6O 12A ··· 12H 12I 12J 12K 12L 12M 12N 12O 12P order 1 2 2 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 6 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 3 3 9 9 27 27 2 2 2 4 4 4 8 3 3 3 3 9 9 9 9 2 2 2 4 4 4 6 6 6 6 8 12 12 18 18 6 ··· 6 12 12 12 12 18 18 18 18

54 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 4 8 8 type + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 S3 S3 D6 D6 D6 D6 C4×S3 C4×S3 S32 S32 C6.D6 C2×S32 C4×S32 S33 S3×C6.D6 kernel S3×C6.D6 C3×S3×Dic3 C3×C6.D6 C33⋊8(C2×C4) C33⋊9(C2×C4) C2×S3×C3⋊S3 S3×C3⋊S3 S3×Dic3 C6.D6 C3×Dic3 C3⋊Dic3 S3×C6 C2×C3⋊S3 C3×S3 C3⋊S3 Dic3 D6 S3 C6 C3 C2 C1 # reps 1 2 1 2 1 1 8 2 1 4 2 2 1 8 4 2 1 2 3 4 1 1

Matrix representation of S3×C6.D6 in GL6(𝔽13)

 12 1 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 12 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 12 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 5 0 0 0 0 5 0 0 0 0 0 0 0 0 12 0 0 0 0 1 12
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 12 0 0 0 0 0 0 0 1 12 0 0 0 0 0 12

G:=sub<GL(6,GF(13))| [12,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,5,0,0,0,0,5,0,0,0,0,0,0,0,0,1,0,0,0,0,12,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,12,12] >;

S3×C6.D6 in GAP, Magma, Sage, TeX

S_3\times C_6.D_6
% in TeX

G:=Group("S3xC6.D6");
// GroupNames label

G:=SmallGroup(432,595);
// by ID

G=gap.SmallGroup(432,595);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,64,58,298,2028,14118]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^2=c^6=e^2=1,d^6=c^3,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=e*c*e=c^-1,e*d*e=d^5>;
// generators/relations

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