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G = D6.3S32  order 432 = 24·33

3rd non-split extension by D6 of S32 acting via S32/C3xS3=C2

metabelian, supersoluble, monomial

Aliases: D6.3S32, Dic3.3S32, C3:D12:5S3, (S3xDic3):3S3, (S3xC6).21D6, C32:2Q8:4S3, C33:4(C4oD4), C33:8D4:5C2, C33:7D4:6C2, C3:Dic3.30D6, C3:1(D12:S3), C3:4(D6.3D6), C3:2(D6.6D6), C32:8(C4oD12), (C3xDic3).11D6, C32:9(Q8:3S3), (C32xC6).16C23, C32:10(D4:2S3), (C32xDic3).5C22, C2.16S33, C6.16(C2xS32), (C3xS3xDic3):5C2, C33:8(C2xC4):3C2, C33:9(C2xC4):3C2, (C2xC3:S3).18D6, (S3xC3xC6).7C22, (C3xC3:D12):7C2, (C3xC32:2Q8):5C2, (C6xC3:S3).21C22, (C3xC6).65(C22xS3), (C3xC3:Dic3).13C22, (C2xC33:C2).5C22, SmallGroup(432,609)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32xC6 — D6.3S32
Chief series
C1 — C3 — C32 — C33 — C32xC6 — S3xC3xC6 — C3xS3xDic3 — D6.3S32
Lower central
C33 — C32xC6 — D6.3S32
Upper central
C1 — C2

Generators and relations for D6.3S32
 G = < a,b,c,d,e,f | a3=b2=c6=e3=f2=1, d2=c3, bab=faf=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, fbf=bc3, dcd-1=c-1, ce=ec, cf=fc, de=ed, fdf=c3d, fef=e-1 >

Subgroups: 1436 in 218 conjugacy classes, 46 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C2xC4, D4, Q8, C32, C32, Dic3, Dic3, C12, D6, D6, C2xC6, C4oD4, C3xS3, C3:S3, C3xC6, C3xC6, Dic6, C4xS3, D12, C2xDic3, C3:D4, C2xC12, C3xD4, C3xQ8, C33, C3xDic3, C3xDic3, C3:Dic3, C3xC12, S3xC6, S3xC6, C2xC3:S3, C2xC3:S3, C62, C4oD12, D4:2S3, Q8:3S3, S3xC32, C3xC3:S3, C33:C2, C32xC6, S3xDic3, S3xDic3, C6.D6, C3:D12, C3:D12, C32:2Q8, C3xDic6, S3xC12, C3xD12, C6xDic3, C3xC3:D4, C4xC3:S3, C12:S3, C32:7D4, C32xDic3, C3xC3:Dic3, S3xC3xC6, C6xC3:S3, C2xC33:C2, D12:S3, D6.6D6, D6.3D6, C3xS3xDic3, C3xC3:D12, C3xC32:2Q8, C33:8(C2xC4), C33:7D4, C33:8D4, C33:9(C2xC4), D6.3S32
Quotients: C1, C2, C22, S3, C23, D6, C4oD4, C22xS3, S32, C4oD12, D4:2S3, Q8:3S3, C2xS32, D12:S3, D6.6D6, D6.3D6, S33, D6.3S32

Permutation representations of D6.3S32
►On 24 points - transitive group 24T1301
Generators in S24
(1 5 3)(2 6 4)(7 11 9)(8 12 10)(13 15 17)(14 16 18)(19 21 23)(20 22 24)

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

(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)

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

(1 16)(2 17)(3 18)(4 13)(5 14)(6 15)(7 24)(8 19)(9 20)(10 21)(11 22)(12 23)
 
G:=sub<Sym(24)| (1,5,3)(2,6,4)(7,11,9)(8,12,10)(13,15,17)(14,16,18)(19,21,23)(20,22,24), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,18)(8,13)(9,14)(10,15)(11,16)(12,17), (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), (1,5,3)(2,6,4)(7,9,11)(8,10,12)(13,15,17)(14,16,18)(19,23,21)(20,24,22), (1,16)(2,17)(3,18)(4,13)(5,14)(6,15)(7,24)(8,19)(9,20)(10,21)(11,22)(12,23)>;
 
G:=Group( (1,5,3)(2,6,4)(7,11,9)(8,12,10)(13,15,17)(14,16,18)(19,21,23)(20,22,24), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,18)(8,13)(9,14)(10,15)(11,16)(12,17), (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), (1,5,3)(2,6,4)(7,9,11)(8,10,12)(13,15,17)(14,16,18)(19,23,21)(20,24,22), (1,16)(2,17)(3,18)(4,13)(5,14)(6,15)(7,24)(8,19)(9,20)(10,21)(11,22)(12,23) );
 
G=PermutationGroup([[(1,5,3),(2,6,4),(7,11,9),(8,12,10),(13,15,17),(14,16,18),(19,21,23),(20,22,24)], [(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,18),(8,13),(9,14),(10,15),(11,16),(12,17)], [(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)], [(1,5,3),(2,6,4),(7,9,11),(8,10,12),(13,15,17),(14,16,18),(19,23,21),(20,24,22)], [(1,16),(2,17),(3,18),(4,13),(5,14),(6,15),(7,24),(8,19),(9,20),(10,21),(11,22),(12,23)]])
 
G:=TransitiveGroup(24,1301);
 

42 conjugacy classes

class 1 2A2B2C2D3A3B3C3D3E3F3G4A4B4C4D4E6A6B6C6D6E6F6G6H6I6J6K6L6M12A12B12C···12I12J12K12L
order122223333333444446666666666666121212···12121212
size11618542224448669918222444668121212366612···12181836

42 irreducible representations

dim111111112222222224444444488
type+++++++++++++++++-+++++
imageC1C2C2C2C2C2C2C2S3S3S3D6D6D6D6C4oD4C4oD12S32S32D4:2S3Q8:3S3C2xS32D12:S3D6.6D6D6.3D6S33D6.3S32
kernelD6.3S32C3xS3xDic3C3xC3:D12C3xC32:2Q8C33:8(C2xC4)C33:7D4C33:8D4C33:9(C2xC4)S3xDic3C3:D12C32:2Q8C3xDic3C3:Dic3S3xC6C2xC3:S3C33C32Dic3D6C32C32C6C3C3C3C2C1
# reps111111111114221242111322211

Matrix representation of D6.3S32 ►in GL8(Z)

-11000000
-10000000
00-110000
00-100000
0000-1100
0000-1000
000000-11
000000-10
,
00000001
00000010
00000-100
0000-1000
000-10000
00-100000
01000000
10000000
,
01000000
-11000000
00010000
00-110000
00001-100
00001000
0000001-1
00000010
,
00001000
00000100
00000010
00000001
-10000000
0-1000000
00-100000
000-10000
,
-11000000
-10000000
000-10000
001-10000
0000-1100
0000-1000
0000000-1
0000001-1
,
00000-100
0000-1000
0000000-1
000000-10
0-1000000
-10000000
000-10000
00-100000

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

D6.3S32 in GAP, Magma, Sage, TeX 

D_6._3S_3^2
 
% in TeX
 
G:=Group("D6.3S3^2");
 
// GroupNames label
 
G:=SmallGroup(432,609);
 
// by ID
 
G=gap.SmallGroup(432,609);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,64,254,135,58,298,2028,14118]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^3=b^2=c^6=e^3=f^2=1,d^2=c^3,b*a*b=f*a*f=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,f*b*f=b*c^3,d*c*d^-1=c^-1,c*e=e*c,c*f=f*c,d*e=e*d,f*d*f=c^3*d,f*e*f=e^-1>;
 
// generators/relations
 

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