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## G = (S3×C6).D6order 432 = 24·33

### 9th non-split extension by S3×C6 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

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

Generators and relations for (S3×C6).D6
G = < a,b,c,d,e,f | a6=b2=c3=e3=f2=1, d2=a3, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf=a3b, dcd-1=c-1, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >

Subgroups: 1396 in 218 conjugacy classes, 46 normal (18 characteristic)
C1, C2, C2, C3, C3, C3, C4, C22, S3, C6, C6, C6, C2×C4, D4, Q8, C32, C32, C32, Dic3, Dic3, C12, D6, D6, C2×C6, C4○D4, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C33, C3×Dic3, C3×Dic3, C3⋊Dic3, C3⋊Dic3, C3×C12, S3×C6, S3×C6, C2×C3⋊S3, C62, C4○D12, D42S3, S3×C32, C33⋊C2, C32×C6, S3×Dic3, C6.D6, D6⋊S3, C3⋊D12, C322Q8, S3×C12, C6×Dic3, C3×C3⋊D4, C4×C3⋊S3, C327D4, C32×Dic3, C3×C3⋊Dic3, C3×C3⋊Dic3, S3×C3×C6, C2×C33⋊C2, D6.D6, D6.3D6, C3×S3×Dic3, C3×D6⋊S3, C338(C2×C4), C337D4, C335Q8, (S3×C6).D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C22×S3, S32, C4○D12, D42S3, C2×S32, D6.D6, D6.3D6, S33, (S3×C6).D6

Permutation representations of (S3×C6).D6
On 24 points - transitive group 24T1303
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 12)(2 11)(3 10)(4 9)(5 8)(6 7)(13 22)(14 21)(15 20)(16 19)(17 24)(18 23)
(1 3 5)(2 4 6)(7 11 9)(8 12 10)(13 17 15)(14 18 16)(19 21 23)(20 22 24)
(1 14 4 17)(2 15 5 18)(3 16 6 13)(7 22 10 19)(8 23 11 20)(9 24 12 21)
(1 3 5)(2 4 6)(7 11 9)(8 12 10)(13 15 17)(14 16 18)(19 23 21)(20 24 22)
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 14)(8 15)(9 16)(10 17)(11 18)(12 13)

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

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

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

G:=TransitiveGroup(24,1303);

45 conjugacy classes

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

45 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 4 4 8 8 type + + + + + + + + + + + + + - + + + image C1 C2 C2 C2 C2 C2 S3 S3 D6 D6 D6 C4○D4 C4○D12 S32 S32 D4⋊2S3 C2×S32 D6.D6 D6.3D6 S33 (S3×C6).D6 kernel (S3×C6).D6 C3×S3×Dic3 C3×D6⋊S3 C33⋊8(C2×C4) C33⋊7D4 C33⋊5Q8 S3×Dic3 D6⋊S3 C3×Dic3 C3⋊Dic3 S3×C6 C33 C32 Dic3 D6 C32 C6 C3 C3 C2 C1 # reps 1 2 1 1 2 1 2 1 2 3 4 2 8 1 2 1 3 2 4 1 1

Matrix representation of (S3×C6).D6 in GL8(ℤ)

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

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

(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(432,606);
// by ID

G=gap.SmallGroup(432,606);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,64,254,298,2028,14118]);
// Polycyclic

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

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