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## G = S3×C3⋊D12order 432 = 24·33

### Direct product of S3 and C3⋊D12

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32×C6 — S3×C3⋊D12
 Chief series C1 — C3 — C32 — C33 — C32×C6 — S3×C3×C6 — S32×C6 — S3×C3⋊D12
 Lower central C33 — C32×C6 — S3×C3⋊D12
 Upper central C1 — C2

Generators and relations for S3×C3⋊D12
G = < a,b,c,d,e | a3=b2=c3=d12=e2=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=ece=c-1, ede=d-1 >

Subgroups: 2052 in 290 conjugacy classes, 54 normal (46 characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, S3, C6, C6, C2×C4, D4, C23, C32, C32, Dic3, Dic3, C12, D6, D6, C2×C6, C2×D4, C3×S3, C3×S3, C3⋊S3, C3×C6, C3×C6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×C6, C33, C3×Dic3, C3×Dic3, C3⋊Dic3, C3×C12, S32, S3×C6, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C62, C2×D12, S3×D4, C2×C3⋊D4, S3×C32, S3×C32, C3×C3⋊S3, C33⋊C2, C32×C6, S3×Dic3, D6⋊S3, C3⋊D12, C3⋊D12, S3×C12, C3×D12, C6×Dic3, C3×C3⋊D4, C12⋊S3, C327D4, C2×S32, C2×S32, S3×C2×C6, C22×C3⋊S3, C32×Dic3, C3×C3⋊Dic3, C3×S32, S3×C3⋊S3, S3×C3×C6, C6×C3⋊S3, C2×C33⋊C2, S3×D12, C2×C3⋊D12, S3×C3⋊D4, C3×S3×Dic3, C3×C3⋊D12, C337D4, C338D4, C339D4, S32×C6, C2×S3×C3⋊S3, S3×C3⋊D12
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D12, C3⋊D4, C22×S3, S32, C2×D12, S3×D4, C2×C3⋊D4, C3⋊D12, C2×S32, S3×D12, C2×C3⋊D12, S3×C3⋊D4, S33, S3×C3⋊D12

Permutation representations of S3×C3⋊D12
On 24 points - transitive group 24T1295
Generators in S24
(1 9 5)(2 10 6)(3 11 7)(4 12 8)(13 17 21)(14 18 22)(15 19 23)(16 20 24)
(1 16)(2 17)(3 18)(4 19)(5 20)(6 21)(7 22)(8 23)(9 24)(10 13)(11 14)(12 15)
(1 5 9)(2 10 6)(3 7 11)(4 12 8)(13 21 17)(14 18 22)(15 23 19)(16 20 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 18)(2 17)(3 16)(4 15)(5 14)(6 13)(7 24)(8 23)(9 22)(10 21)(11 20)(12 19)

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

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

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

G:=TransitiveGroup(24,1295);

45 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 3E 3F 3G 4A 4B 6A 6B 6C 6D 6E 6F 6G ··· 6L 6M 6N ··· 6R 6S 6T 6U 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 12 12 12 12 12 12 12 size 1 1 3 3 6 18 18 54 2 2 2 4 4 4 8 6 18 2 2 2 4 4 4 6 ··· 6 8 12 ··· 12 18 18 36 6 6 12 12 12 18 18

45 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 8 8 type + + + + + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 S3 S3 S3 D4 D6 D6 D6 D6 D12 C3⋊D4 S32 S32 S3×D4 C3⋊D12 C2×S32 S3×D12 S3×C3⋊D4 S33 S3×C3⋊D12 kernel S3×C3⋊D12 C3×S3×Dic3 C3×C3⋊D12 C33⋊7D4 C33⋊8D4 C33⋊9D4 S32×C6 C2×S3×C3⋊S3 S3×Dic3 C3⋊D12 C2×S32 S3×C32 C3×Dic3 C3⋊Dic3 S3×C6 C2×C3⋊S3 C3×S3 C3×S3 Dic3 D6 C32 S3 C6 C3 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 1 1 1 2 2 1 4 2 4 4 1 2 1 2 3 2 2 1 1

Matrix representation of S3×C3⋊D12 in GL8(ℤ)

 -1 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 1 0 0 0 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 0 0 0 -1 -1 0 0 0 0 1 0 0 0 0 0 0 0 -1 -1 0 0 0 0 1 0 0 0 0 0 0 0 -1 -1 0 0 0 0 1 0 0 0 0 0 0 0 -1 -1 0 0 0 0 0 0
,
 -1 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 -1
,
 0 0 0 1 0 0 0 0 0 0 -1 -1 0 0 0 0 0 -1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 -1 0 0 0 0 0 -1 -1 0 0 0 0 0 0 1 0 0 0
,
 0 0 0 0 0 0 1 1 0 0 0 0 0 0 -1 0 0 0 0 0 -1 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 -1 0 0 0 0 0 -1 0 0 0 0 0 0 1 1 0 0 0 0 0 0

G:=sub<GL(8,Integers())| [-1,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,0],[0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0],[-1,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,-1],[0,0,0,1,0,0,0,0,0,0,-1,1,0,0,0,0,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,0,0,0,0,0,1,-1,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,1,0,0,0,0,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,-1,1,0,0,0,0,0,0,-1,0,0,0,0,0,1,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0] >;

S3×C3⋊D12 in GAP, Magma, Sage, TeX

S_3\times C_3\rtimes D_{12}
% in TeX

G:=Group("S3xC3:D12");
// GroupNames label

G:=SmallGroup(432,598);
// by ID

G=gap.SmallGroup(432,598);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,135,58,298,2028,14118]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^2=c^3=d^12=e^2=1,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^-1>;
// generators/relations

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