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G = (S3×C6)⋊D6  order 432 = 24·33

1st semidirect product of S3×C6 and D6 acting via D6/C3=C22

metabelian, supersoluble, monomial

Aliases: D6⋊5S32, (S3×C6)⋊1D6, Dic3⋊4S32, C33⋊9(C2×D4), C32⋊8(S3×D4), C3⋊D12⋊4S3, D6⋊S3⋊3S3, C3⋊Dic3⋊14D6, (C3×Dic3)⋊2D6, C33⋊9D4⋊8C2, C33⋊C2⋊2D4, C3⋊1(D6⋊D6), C3⋊2(Dic3⋊D6), (C32×C6).8C23, (C32×Dic3)⋊3C22, C2.8S33, C6.8(C2×S32), (C2×C3⋊S3)⋊13D6, (S3×C3×C6)⋊5C22, C33⋊8(C2×C4)⋊1C2, (C6×C3⋊S3)⋊4C22, (C3×C3⋊D12)⋊3C2, (C3×D6⋊S3)⋊4C2, (C3×C6).57(C22×S3), (C3×C3⋊Dic3)⋊4C22, (C2×C33⋊C2).2C22, (C2×S3×C3⋊S3)⋊3C2, SmallGroup(432,601)

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×D6⋊S3 — (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 | a6=b3=c2=d6=e2=1, ab=ba, ac=ca, dad-1=eae=a-1, cbc=ebe=b-1, bd=db, dcd-1=a3c, ce=ec, ede=d-1 >

Subgroups: 2276 in 306 conjugacy classes, 48 normal (18 characteristic)
C1, C2, C2, C3, C3, C3, C4, C22, S3, C6, C6, C6, C2×C4, D4, C23, C32, C32, C32, Dic3, Dic3, C12, D6, D6, C2×C6, C2×D4, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, C4×S3, D12, C3⋊D4, C3×D4, C22×S3, C33, C3×Dic3, C3×Dic3, C3⋊Dic3, C3×C12, S32, S3×C6, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C62, S3×D4, S3×C32, C3×C3⋊S3, C33⋊C2, C32×C6, C6.D6, D6⋊S3, D6⋊S3, C3⋊D12, C3⋊D12, C3×D12, C3×C3⋊D4, C4×C3⋊S3, C2×S32, C22×C3⋊S3, C32×Dic3, C3×C3⋊Dic3, S3×C3⋊S3, S3×C3×C6, C6×C3⋊S3, C2×C33⋊C2, D6⋊D6, Dic3⋊D6, C3×D6⋊S3, C3×C3⋊D12, C33⋊8(C2×C4), C33⋊9D4, C2×S3×C3⋊S3, (S3×C6)⋊D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C22×S3, S32, S3×D4, C2×S32, D6⋊D6, Dic3⋊D6, S33, (S3×C6)⋊D6

Permutation representations of (S3×C6)⋊D6
►On 24 points - transitive group 24T1294
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 3 5)(2 4 6)(7 9 11)(8 10 12)(13 17 15)(14 18 16)(19 23 21)(20 24 22)

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

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

(1 3)(4 6)(8 12)(9 11)(13 17)(14 16)(20 24)(21 23)
 
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), (1,3,5)(2,4,6)(7,9,11)(8,10,12)(13,17,15)(14,18,16)(19,23,21)(20,24,22), (1,17)(2,18)(3,13)(4,14)(5,15)(6,16)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24), (1,19,3,23,5,21)(2,24,4,22,6,20)(7,16,11,18,9,14)(8,15,12,17,10,13), (1,3)(4,6)(8,12)(9,11)(13,17)(14,16)(20,24)(21,23)>;
 
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), (1,3,5)(2,4,6)(7,9,11)(8,10,12)(13,17,15)(14,18,16)(19,23,21)(20,24,22), (1,17)(2,18)(3,13)(4,14)(5,15)(6,16)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24), (1,19,3,23,5,21)(2,24,4,22,6,20)(7,16,11,18,9,14)(8,15,12,17,10,13), (1,3)(4,6)(8,12)(9,11)(13,17)(14,16)(20,24)(21,23) );
 
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)], [(1,3,5),(2,4,6),(7,9,11),(8,10,12),(13,17,15),(14,18,16),(19,23,21),(20,24,22)], [(1,17),(2,18),(3,13),(4,14),(5,15),(6,16),(7,19),(8,20),(9,21),(10,22),(11,23),(12,24)], [(1,19,3,23,5,21),(2,24,4,22,6,20),(7,16,11,18,9,14),(8,15,12,17,10,13)], [(1,3),(4,6),(8,12),(9,11),(13,17),(14,16),(20,24),(21,23)]])
 
G:=TransitiveGroup(24,1294);
 

39 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C3D3E3F3G4A4B6A6B6C6D6E6F6G6H···6O6P6Q12A12B12C12D12E
order1222222233333334466666666···6661212121212
size1166181827272224448618222444812···1236361212121236

39 irreducible representations

dim111111222222244444488
type++++++++++++++++++++
imageC1C2C2C2C2C2S3S3D4D6D6D6D6S32S32S3×D4C2×S32D6⋊D6Dic3⋊D6S33(S3×C6)⋊D6
kernel(S3×C6)⋊D6C3×D6⋊S3C3×C3⋊D12C33⋊8(C2×C4)C33⋊9D4C2×S3×C3⋊S3D6⋊S3C3⋊D12C33⋊C2C3×Dic3C3⋊Dic3S3×C6C2×C3⋊S3Dic3D6C32C6C3C3C2C1
# reps112112122214212332411

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

0-1000000
11000000
000-10000
00110000
00000-100
00001100
0000000-1
00000011
,
-1-1000000
10000000
00-1-10000
00100000
00000100
0000-1-100
00000001
000000-1-1
,
00001000
00000100
00000010
00000001
10000000
01000000
00100000
00010000
,
00000011
0000000-1
00000100
00001000
00-1-10000
00010000
0-1000000
-10000000
,
0-1000000
-10000000
00110000
000-10000
00000-100
0000-1000
00000011
0000000-1

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

(S3×C6)⋊D6 in GAP, Magma, Sage, TeX 

(S_3\times C_6)\rtimes D_6
 
% in TeX
 
G:=Group("(S3xC6):D6");
 
// GroupNames label
 
G:=SmallGroup(432,601);
 
// by ID
 
G=gap.SmallGroup(432,601);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,141,254,135,58,298,2028,14118]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^6=b^3=c^2=d^6=e^2=1,a*b=b*a,a*c=c*a,d*a*d^-1=e*a*e=a^-1,c*b*c=e*b*e=b^-1,b*d=d*b,d*c*d^-1=a^3*c,c*e=e*c,e*d*e=d^-1>;
 
// generators/relations
 

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