Copied to
clipboard

G = (S3xC6).D6  order 432 = 24·33

9th non-split extension by S3xC6 of D6 acting via D6/C3=C22

metabelian, supersoluble, monomial

Aliases: D6.1S32, (S3xC6).9D6, Dic3.8S32, D6:S3:6S3, (S3xDic3):4S3, C33:1(C4oD4), C33:7D4:2C2, C33:5Q8:5C2, C3:Dic3.29D6, C3:1(D6.3D6), C3:1(D6.D6), (C3xDic3).23D6, C32:15(C4oD12), C32:7(D4:2S3), (C32xC6).13C23, (C32xDic3).19C22, C2.13S33, C6.13(C2xS32), (C3xS3xDic3):1C2, C33:8(C2xC4):2C2, (S3xC3xC6).4C22, (C3xD6:S3):5C2, (C3xC6).62(C22xS3), (C3xC3:Dic3).5C22, (C2xC33:C2).4C22, SmallGroup(432,606)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32xC6 — (S3xC6).D6
Chief series
C1 — C3 — C32 — C33 — C32xC6 — S3xC3xC6 — C3xS3xDic3 — (S3xC6).D6
Lower central
C33 — C32xC6 — (S3xC6).D6
Upper central
C1 — C2

Generators and relations for (S3xC6).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, C2xC4, D4, Q8, C32, C32, C32, Dic3, Dic3, C12, D6, D6, C2xC6, C4oD4, C3xS3, C3:S3, C3xC6, C3xC6, C3xC6, Dic6, C4xS3, D12, C2xDic3, C3:D4, C2xC12, C3xD4, C33, C3xDic3, C3xDic3, C3:Dic3, C3:Dic3, C3xC12, S3xC6, S3xC6, C2xC3:S3, C62, C4oD12, D4:2S3, S3xC32, C33:C2, C32xC6, S3xDic3, C6.D6, D6:S3, C3:D12, C32:2Q8, S3xC12, C6xDic3, C3xC3:D4, C4xC3:S3, C32:7D4, C32xDic3, C3xC3:Dic3, C3xC3:Dic3, S3xC3xC6, C2xC33:C2, D6.D6, D6.3D6, C3xS3xDic3, C3xD6:S3, C33:8(C2xC4), C33:7D4, C33:5Q8, (S3xC6).D6
Quotients: C1, C2, C22, S3, C23, D6, C4oD4, C22xS3, S32, C4oD12, D4:2S3, C2xS32, D6.D6, D6.3D6, S33, (S3xC6).D6

Permutation representations of (S3xC6).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+++++++++++++-+++
imageC1C2C2C2C2C2S3S3D6D6D6C4oD4C4oD12S32S32D4:2S3C2xS32D6.D6D6.3D6S33(S3xC6).D6
kernel(S3xC6).D6C3xS3xDic3C3xD6:S3C33:8(C2xC4)C33:7D4C33:5Q8S3xDic3D6:S3C3xDic3C3:Dic3S3xC6C33C32Dic3D6C32C6C3C3C2C1
# reps121121212342812132411

Matrix representation of (S3xC6).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] >;
 

(S3xC6).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
 

׿
x
:
Z
F
o
wr
Q
<