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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

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