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G = (S3×C6).D6  order 432 = 24·33

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

metabelian, supersoluble, monomial

Aliases: D6.1S32, (S3×C6).9D6, Dic3.8S32, D6⋊S3⋊6S3, (S3×Dic3)⋊4S3, C33⋊1(C4○D4), C33⋊7D4⋊2C2, C33⋊5Q8⋊5C2, C3⋊Dic3.29D6, C3⋊1(D6.3D6), C3⋊1(D6.D6), (C3×Dic3).23D6, C32⋊15(C4○D12), C32⋊7(D4⋊2S3), (C32×C6).13C23, (C32×Dic3).19C22, C2.13S33, C6.13(C2×S32), (C3×S3×Dic3)⋊1C2, C33⋊8(C2×C4)⋊2C2, (S3×C3×C6).4C22, (C3×D6⋊S3)⋊5C2, (C3×C6).62(C22×S3), (C3×C3⋊Dic3).5C22, (C2×C33⋊C2).4C22, SmallGroup(432,606)

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, D4⋊2S3, S3×C32, C33⋊C2, C32×C6, S3×Dic3, C6.D6, D6⋊S3, C3⋊D12, C32⋊2Q8, S3×C12, C6×Dic3, C3×C3⋊D4, C4×C3⋊S3, C32⋊7D4, C32×Dic3, C3×C3⋊Dic3, C3×C3⋊Dic3, S3×C3×C6, C2×C33⋊C2, D6.D6, D6.3D6, C3×S3×Dic3, C3×D6⋊S3, C33⋊8(C2×C4), C33⋊7D4, C33⋊5Q8, (S3×C6).D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C22×S3, S32, C4○D12, D4⋊2S3, 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 2A2B2C2D3A3B3C3D3E3F3G4A4B4C4D4E6A6B6C6D6E6F6G6H6I6J6K6L···6Q12A12B12C12D12E12F12G12H12I12J12K
order12222333333344444666666666666···61212121212121212121212
size1166542224448331818182224446666812···12666612121818181836

45 irreducible representations

dim111111222222244444488
type+++++++++++++-+++
imageC1C2C2C2C2C2S3S3D6D6D6C4○D4C4○D12S32S32D4⋊2S3C2×S32D6.D6D6.3D6S33(S3×C6).D6
kernel(S3×C6).D6C3×S3×Dic3C3×D6⋊S3C33⋊8(C2×C4)C33⋊7D4C33⋊5Q8S3×Dic3D6⋊S3C3×Dic3C3⋊Dic3S3×C6C33C32Dic3D6C32C6C3C3C2C1
# reps121121212342812132411

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

01000000
-11000000
001-10000
00100000
00000100
0000-1100
0000001-1
00000010
,
001-10000
00100000
01000000
-11000000
0000001-1
00000010
00000100
0000-1100
,
0-1000000
1-1000000
000-10000
001-10000
0000-1100
0000-1000
000000-11
000000-10
,
00001000
00000100
00000010
00000001
-10000000
0-1000000
00-100000
000-10000
,
0-1000000
1-1000000
000-10000
001-10000
00000-100
00001-100
0000000-1
0000001-1
,
00000001
00000010
00000-100
0000-1000
000-10000
00-100000
01000000
10000000

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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