metabelian, supersoluble, monomial
Aliases: D6.1(S32), (S3×C6).9D6, D6⋊S3⋊6S3, (S3×Dic3)⋊4S3, C33⋊1(C4○D4), C33⋊7D4⋊2C2, C33⋊5Q8⋊5C2, Dic3.8(S32), C3⋊Dic3.29D6, C3⋊1(D6.D6), C3⋊1(D6.3D6), (C3×Dic3).23D6, C32⋊15(C4○D12), C32⋊7(D4⋊2S3), (C32×C6).13C23, (C32×Dic3).19C22, C2.13(S33), C6.13(C2×S32), (C3×S3×Dic3)⋊1C2, (S3×C3×C6).4C22, (C3×D6⋊S3)⋊5C2, C33⋊8(C2×C4)⋊2C2, (C3×C6).62(C22×S3), (C3×C3⋊Dic3).5C22, (C2×C33⋊C2).4C22, SmallGroup(432,606)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D6.(S32)
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 [×3], C3, C3 [×2], C3 [×4], C4 [×4], C22 [×3], S3 [×11], C6, C6 [×2], C6 [×10], C2×C4 [×3], D4 [×3], Q8, C32, C32 [×2], C32 [×4], Dic3, Dic3 [×9], C12 [×6], D6 [×2], D6 [×7], C2×C6 [×6], C4○D4, C3×S3 [×6], C3⋊S3 [×9], C3×C6, C3×C6 [×2], C3×C6 [×6], Dic6 [×3], C4×S3 [×6], D12 [×2], C2×Dic3 [×2], C3⋊D4 [×8], C2×C12 [×2], C3×D4, C33, C3×Dic3 [×2], C3×Dic3 [×10], C3⋊Dic3, C3⋊Dic3 [×2], C3×C12, S3×C6 [×4], S3×C6 [×2], C2×C3⋊S3 [×7], C62 [×2], C4○D12 [×2], D4⋊2S3, S3×C32 [×2], C33⋊C2, C32×C6, S3×Dic3 [×2], C6.D6 [×3], D6⋊S3, C3⋊D12 [×6], C32⋊2Q8 [×3], S3×C12 [×2], C6×Dic3 [×2], C3×C3⋊D4 [×2], C4×C3⋊S3, C32⋊7D4 [×2], C32×Dic3, C3×C3⋊Dic3, C3×C3⋊Dic3 [×2], S3×C3×C6 [×2], C2×C33⋊C2, D6.D6, D6.3D6 [×2], C3×S3×Dic3 [×2], C3×D6⋊S3, C33⋊8(C2×C4), C33⋊7D4 [×2], C33⋊5Q8, D6.(S32)
Quotients: C1, C2 [×7], C22 [×7], S3 [×3], C23, D6 [×9], C4○D4, C22×S3 [×3], S32 [×3], C4○D12 [×2], D4⋊2S3, C2×S32 [×3], D6.D6, D6.3D6 [×2], S33, D6.(S32)
(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 | D6.(S32) |
kernel | D6.(S32) | 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 D6.(S32) ►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] >;
D6.(S32) in GAP, Magma, Sage, TeX
D_6.(S_3^2)
% in TeX
G:=Group("D6.(S3^2)");
// 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