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G = S3×2+ 1+4  order 192 = 26·3

Direct product of S3 and 2+ 1+4

direct product, metabelian, supersoluble, monomial, rational, 2-hyperelementary

Aliases: S3×2+ 1+4, C6.15C25, D12⋊13C23, C12.50C24, D6.14C24, Dic6⋊12C23, Dic3.10C24, C4○D4⋊13D6, (C2×D4)⋊31D6, D4○D12⋊11C2, D4⋊6D6⋊9C2, (C4×S3)⋊3C23, (C2×C12)⋊2C23, C3⋊D4⋊6C23, (C2×C6).6C24, D4⋊10(C22×S3), (C6×D4)⋊25C22, (C3×D4)⋊11C23, (S3×D4)⋊17C22, C4.47(S3×C23), C2.16(S3×C24), C23⋊3(C22×S3), (C22×C6)⋊2C23, Q8⋊10(C22×S3), (C3×Q8)⋊10C23, (S3×Q8)⋊20C22, C3⋊3(C2×2+ 1+4), C4○D12⋊12C22, (C2×D12)⋊40C22, (C22×S3)⋊6C23, (C2×Dic3)⋊6C23, C22.3(S3×C23), D4⋊2S3⋊15C22, (S3×C23)⋊19C22, Q8⋊3S3⋊15C22, (C3×2+ 1+4)⋊4C2, (C2×S3×D4)⋊29C2, (S3×C4○D4)⋊7C2, (S3×C2×C4)⋊36C22, (C2×C4)⋊2(C22×S3), (C3×C4○D4)⋊10C22, (C2×C3⋊D4)⋊32C22, SmallGroup(192,1524)

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, D4⋊2S3, S3×Q8, Q8⋊3S3, C2×C3⋊D4, C6×D4, C3×C4○D4, S3×C23, C2×2+ 1+4, C2×S3×D4, D4⋊6D6, 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 2A2B···2J2K2L2M···2U 3 4A···4F4G···4L6A6B···6J12A···12F
order122···2222···234···44···466···612···12
size112···2336···622···26···624···44···4

51 irreducible representations

dim11111122248
type+++++++++++
imageC1C2C2C2C2C2S3D6D62+ 1+4S3×2+ 1+4
kernelS3×2+ 1+4C2×S3×D4D4⋊6D6S3×C4○D4D4○D12C3×2+ 1+42+ 1+4C2×D4C4○D4S3C1
# reps19966119621

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

-1-10000
100000
001000
000100
000010
000001
,
-100000
110000
001000
000100
000010
000001
,
-100000
0-10000
00-1200
00-1100
000-10-1
001-110
,
-100000
0-10000
001000
001-100
000010
00-100-1
,
100000
010000
001020
00001-1
00-10-10
00-11-10
,
100000
010000
001000
000100
00-10-10
00-100-1

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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