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G = 2+ 1+46S3order 192 = 26·3

1st semidirect product of 2+ 1+4 and S3 acting via S3/C3=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: 2+ 1+46S3, (C3×D4)⋊15D4, (C3×Q8)⋊15D4, D4⋊D65C2, C123D48C2, C35(D44D4), D47(C3⋊D4), C4○D4.25D6, Q88(C3⋊D4), (C2×D4).82D6, C6.80C22≀C2, C12.217(C2×D4), (C22×C6).24D4, C12.D411C2, (C2×D12)⋊15C22, (C2×C12).21C23, (C4×Dic3)⋊8C22, Q83Dic311C2, (C6×D4).107C22, C23.12(C3⋊D4), C4.Dic310C22, (C3×2+ 1+4)⋊1C2, C2.14(C244S3), (C2×C6).42(C2×D4), C4.64(C2×C3⋊D4), (C2×C4).21(C22×S3), C22.14(C2×C3⋊D4), (C3×C4○D4).19C22, SmallGroup(192,800)

Series: Derived Chief Lower central Upper central

C1C2×C12 — 2+ 1+46S3
C1C3C6C2×C6C2×C12C2×D12D4⋊D6 — 2+ 1+46S3
C3C6C2×C12 — 2+ 1+46S3
C1C2C2×C42+ 1+4

Generators and relations for 2+ 1+46S3
 G = < a,b,c,d,e,f | a4=b2=d2=e3=f2=1, c2=a2, bab=faf=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, fbf=ab, dcd=fcf=a2c, ce=ec, de=ed, fdf=cd, fef=e-1 >

Subgroups: 520 in 168 conjugacy classes, 43 normal (17 characteristic)
C1, C2, C2 [×6], C3, C4 [×2], C4 [×4], C22, C22 [×11], S3, C6, C6 [×5], C8 [×2], C2×C4, C2×C4 [×5], D4 [×2], D4 [×14], Q8 [×2], C23 [×2], C23 [×3], Dic3 [×2], C12 [×2], C12 [×2], D6 [×3], C2×C6, C2×C6 [×8], C42, M4(2) [×2], D8 [×2], SD16 [×2], C2×D4, C2×D4 [×7], C4○D4 [×2], C4○D4 [×2], C3⋊C8 [×2], D12 [×2], C2×Dic3, C3⋊D4 [×4], C2×C12, C2×C12 [×4], C3×D4 [×2], C3×D4 [×8], C3×Q8 [×2], C22×S3, C22×C6 [×2], C22×C6 [×2], C4.D4, C4≀C2 [×2], C41D4, C8⋊C22 [×2], 2+ 1+4, C4.Dic3 [×2], C4×Dic3, D4⋊S3 [×2], Q82S3 [×2], C2×D12, C2×C3⋊D4 [×2], C6×D4, C6×D4 [×4], C3×C4○D4 [×2], C3×C4○D4 [×2], D44D4, C12.D4, Q83Dic3 [×2], C123D4, D4⋊D6 [×2], C3×2+ 1+4, 2+ 1+46S3
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×6], C23, D6 [×3], C2×D4 [×3], C3⋊D4 [×6], C22×S3, C22≀C2, C2×C3⋊D4 [×3], D44D4, C244S3, 2+ 1+46S3

Permutation representations of 2+ 1+46S3
On 24 points - transitive group 24T347
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 24)(2 23)(3 22)(4 21)(5 13)(6 16)(7 15)(8 14)(9 19)(10 18)(11 17)(12 20)
(1 2 3 4)(5 8 7 6)(9 12 11 10)(13 14 15 16)(17 18 19 20)(21 24 23 22)
(1 23)(2 24)(3 21)(4 22)(5 16)(6 13)(7 14)(8 15)(9 20)(10 17)(11 18)(12 19)
(1 16 17)(2 13 18)(3 14 19)(4 15 20)(5 10 23)(6 11 24)(7 12 21)(8 9 22)
(2 4)(5 11)(6 10)(7 9)(8 12)(13 20)(14 19)(15 18)(16 17)(21 22)(23 24)

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

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

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

G:=TransitiveGroup(24,347);

33 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B4C4D4E4F6A6B···6J8A8B12A···12F
order12222222344444466···68812···12
size11244442422244121224···424244···4

33 irreducible representations

dim11111122222222248
type++++++++++++++
imageC1C2C2C2C2C2S3D4D4D4D6D6C3⋊D4C3⋊D4C3⋊D4D44D42+ 1+46S3
kernel2+ 1+46S3C12.D4Q83Dic3C123D4D4⋊D6C3×2+ 1+42+ 1+4C3×D4C3×Q8C22×C6C2×D4C4○D4D4Q8C23C3C1
# reps11212112221244421

Matrix representation of 2+ 1+46S3 in GL6(𝔽73)

7200000
0720000
0072200
0072100
0000722
0000721
,
1720000
0720000
0000722
000001
0072200
000100
,
100000
010000
0017100
0017200
0000722
0000721
,
7200000
0720000
0000722
0000721
0017100
0017200
,
64450000
080000
001000
000100
000010
000001
,
100000
2720000
001000
0017200
0000171
0000072

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,72,0,0,0,0,2,1,0,0,0,0,0,0,72,72,0,0,0,0,2,1],[1,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,2,1,0,0,72,0,0,0,0,0,2,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,71,72,0,0,0,0,0,0,72,72,0,0,0,0,2,1],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,1,0,0,0,0,71,72,0,0,72,72,0,0,0,0,2,1,0,0],[64,0,0,0,0,0,45,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,2,0,0,0,0,0,72,0,0,0,0,0,0,1,1,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,71,72] >;

2+ 1+46S3 in GAP, Magma, Sage, TeX

2_+^{1+4}\rtimes_6S_3
% in TeX

G:=Group("ES+(2,2):6S3");
// GroupNames label

G:=SmallGroup(192,800);
// by ID

G=gap.SmallGroup(192,800);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,254,570,1684,851,438,102,6278]);
// Polycyclic

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

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