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## G = S32⋊Dic3order 432 = 24·33

### The semidirect product of S32 and Dic3 acting via Dic3/C6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3×C3⋊S3 — S32⋊Dic3
 Chief series C1 — C3 — C33 — C3×C3⋊S3 — C6×C3⋊S3 — C33⋊9(C2×C4) — S32⋊Dic3
 Lower central C33 — C3×C3⋊S3 — S32⋊Dic3
 Upper central C1 — C2

Generators and relations for S32⋊Dic3
G = < a,b,c,d,e,f | a3=b2=c3=d2=e6=1, f2=e3, bab=a-1, ac=ca, ad=da, ae=ea, faf-1=c, bc=cb, bd=db, be=eb, fbf-1=d, dcd=c-1, ce=ec, fcf-1=a, de=ed, fdf-1=b, fef-1=e-1 >

Subgroups: 804 in 132 conjugacy classes, 23 normal (19 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 [×4], C12, D6 [×7], C2×C6 [×7], C22⋊C4, C3×S3 [×10], C3⋊S3 [×2], C3×C6, C3×C6 [×6], C4×S3, C2×Dic3 [×2], C22×S3, C22×C6, C33, C3×Dic3 [×3], C3⋊Dic3, C32⋊C4, S32 [×2], S32, S3×C6 [×9], C2×C3⋊S3, C62, C6.D4, S3×C32 [×2], C3×C3⋊S3 [×2], C32×C6, S3×Dic3, C6.D6, C2×C32⋊C4, C2×S32, S3×C2×C6, C3×C3⋊Dic3, C33⋊C4, C3×S32 [×2], C3×S32, S3×C3×C6, C6×C3⋊S3, S32⋊C4, C339(C2×C4), C2×C33⋊C4, S32×C6, S32⋊Dic3
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], Dic3 [×2], D6, C22⋊C4, C2×Dic3, C3⋊D4 [×2], C6.D4, S3≀C2, S32⋊C4, C33⋊D4, S32⋊Dic3

Permutation representations of S32⋊Dic3
On 24 points - transitive group 24T1293
Generators in S24
(13 15 17)(14 16 18)(19 23 21)(20 24 22)
(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)
(1 5 3)(2 6 4)(7 9 11)(8 10 12)
(1 11)(2 12)(3 7)(4 8)(5 9)(6 10)
(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 4 13)(2 15 5 18)(3 14 6 17)(7 20 10 23)(8 19 11 22)(9 24 12 21)

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

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

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

G:=TransitiveGroup(24,1293);

36 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 3D 3E 3F 4A 4B 4C 4D 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 6K ··· 6P 6Q 6R 12A 12B order 1 2 2 2 2 2 3 3 3 3 3 3 4 4 4 4 6 6 6 6 6 6 6 6 6 6 6 ··· 6 6 6 12 12 size 1 1 6 6 9 9 2 4 4 4 4 8 18 18 54 54 2 4 4 4 4 6 6 6 6 8 12 ··· 12 18 18 36 36

36 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 4 8 8 type + + + + + + + - + + + + - image C1 C2 C2 C2 C4 S3 D4 D4 Dic3 D6 C3⋊D4 C3⋊D4 S3≀C2 S32⋊C4 S32⋊C4 C33⋊D4 S32⋊Dic3 C33⋊D4 S32⋊Dic3 kernel S32⋊Dic3 C33⋊9(C2×C4) C2×C33⋊C4 S32×C6 C3×S32 C2×S32 C3×C3⋊S3 C32×C6 S32 C2×C3⋊S3 C3⋊S3 C3×C6 C6 C3 C3 C2 C1 C2 C1 # reps 1 1 1 1 4 1 1 1 2 1 2 2 4 2 2 4 4 1 1

Matrix representation of S32⋊Dic3 in GL4(𝔽7) generated by

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

S32⋊Dic3 in GAP, Magma, Sage, TeX

S_3^2\rtimes {\rm Dic}_3
% in TeX

G:=Group("S3^2:Dic3");
// GroupNames label

G:=SmallGroup(432,580);
// by ID

G=gap.SmallGroup(432,580);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,56,85,1684,571,298,677,1027,14118]);
// Polycyclic

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

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