Copied to
clipboard

## G = S3×2+ 1+4order 192 = 26·3

### Direct product of S3 and 2+ 1+4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — S3×2+ 1+4
 Chief series C1 — C3 — C6 — D6 — C22×S3 — S3×C23 — C2×S3×D4 — S3×2+ 1+4
 Lower central C3 — C6 — S3×2+ 1+4
 Upper central C1 — C2 — 2+ 1+4

Generators and relations for S3×2+ 1+4
G = < a,b,c,d,e,f | a3=b2=c4=d2=f2=1, e2=c2, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd=c-1, ce=ec, cf=fc, de=ed, df=fd, fef=c2e >

Subgroups: 2152 in 898 conjugacy classes, 445 normal (8 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, S3, C6, C6, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, Dic3, C12, D6, D6, D6, C2×C6, C2×C6, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, C24, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C3×Q8, C22×S3, C22×S3, C22×C6, C22×D4, C2×C4○D4, 2+ 1+4, 2+ 1+4, S3×C2×C4, C2×D12, C4○D12, S3×D4, D42S3, S3×Q8, Q83S3, C2×C3⋊D4, C6×D4, C3×C4○D4, S3×C23, C2×2+ 1+4, C2×S3×D4, D46D6, S3×C4○D4, D4○D12, C3×2+ 1+4, S3×2+ 1+4
Quotients: C1, C2, C22, S3, C23, D6, C24, C22×S3, 2+ 1+4, C25, S3×C23, C2×2+ 1+4, S3×C24, S3×2+ 1+4

Permutation representations of S3×2+ 1+4
On 24 points - transitive group 24T335
Generators in S24
(1 17 10)(2 18 11)(3 19 12)(4 20 9)(5 21 16)(6 22 13)(7 23 14)(8 24 15)
(1 3)(2 4)(5 7)(6 8)(9 18)(10 19)(11 20)(12 17)(13 24)(14 21)(15 22)(16 23)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(2 4)(6 8)(9 11)(13 15)(18 20)(22 24)
(1 5 3 7)(2 6 4 8)(9 15 11 13)(10 16 12 14)(17 21 19 23)(18 22 20 24)
(1 7)(2 8)(3 5)(4 6)(9 13)(10 14)(11 15)(12 16)(17 23)(18 24)(19 21)(20 22)

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

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

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

G:=TransitiveGroup(24,335);

51 conjugacy classes

 class 1 2A 2B ··· 2J 2K 2L 2M ··· 2U 3 4A ··· 4F 4G ··· 4L 6A 6B ··· 6J 12A ··· 12F order 1 2 2 ··· 2 2 2 2 ··· 2 3 4 ··· 4 4 ··· 4 6 6 ··· 6 12 ··· 12 size 1 1 2 ··· 2 3 3 6 ··· 6 2 2 ··· 2 6 ··· 6 2 4 ··· 4 4 ··· 4

51 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 4 8 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 S3 D6 D6 2+ 1+4 S3×2+ 1+4 kernel S3×2+ 1+4 C2×S3×D4 D4⋊6D6 S3×C4○D4 D4○D12 C3×2+ 1+4 2+ 1+4 C2×D4 C4○D4 S3 C1 # reps 1 9 9 6 6 1 1 9 6 2 1

Matrix representation of S3×2+ 1+4 in GL6(ℤ)

 -1 -1 0 0 0 0 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
,
 -1 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 0 0 0 0 0 0 1
,
 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 2 0 0 0 0 -1 1 0 0 0 0 0 -1 0 -1 0 0 1 -1 1 0
,
 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 1 0 0 0 0 0 1 -1 0 0 0 0 0 0 1 0 0 0 -1 0 0 -1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 2 0 0 0 0 0 1 -1 0 0 -1 0 -1 0 0 0 -1 1 -1 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 -1 0 -1 0 0 0 -1 0 0 -1

G:=sub<GL(6,Integers())| [-1,1,0,0,0,0,-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],[-1,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,-1,0,1,0,0,2,1,-1,-1,0,0,0,0,0,1,0,0,0,0,-1,0],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,1,0,-1,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,-1,-1,0,0,0,0,0,1,0,0,2,1,-1,-1,0,0,0,-1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,-1,-1,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1] >;

S3×2+ 1+4 in GAP, Magma, Sage, TeX

S_3\times 2_+^{1+4}
% in TeX

G:=Group("S3xES+(2,2)");
// GroupNames label

G:=SmallGroup(192,1524);
// by ID

G=gap.SmallGroup(192,1524);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,297,851,6278]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^3=b^2=c^4=d^2=f^2=1,e^2=c^2,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,b*f=f*b,d*c*d=c^-1,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f=c^2*e>;
// generators/relations

׿
×
𝔽