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G = S3×M4(2)  order 96 = 25·3

Direct product of S3 and M4(2)

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

Aliases: S3×M4(2), C86D6, C247C22, Dic3M4(2), C12.38C23, (S3×C8)⋊7C2, C8⋊S35C2, (C4×S3).1C4, C4.15(C4×S3), C3⋊C811C22, D6.6(C2×C4), (C2×C4).45D6, C32(C2×M4(2)), C12.12(C2×C4), C4.Dic35C2, C22.7(C4×S3), (C3×M4(2))⋊5C2, (C22×S3).4C4, C6.15(C22×C4), C4.38(C22×S3), (C2×Dic3).6C4, Dic3.7(C2×C4), (C4×S3).18C22, (C2×C12).25C22, (S3×C2×C4).4C2, C2.16(S3×C2×C4), (C2×C6).5(C2×C4), SmallGroup(96,113)

Series: Derived Chief Lower central Upper central

C1C6 — S3×M4(2)
C1C3C6C12C4×S3S3×C2×C4 — S3×M4(2)
C3C6 — S3×M4(2)
C1C4M4(2)

Generators and relations for S3×M4(2)
 G = < a,b,c,d | a3=b2=c8=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c5 >

Subgroups: 130 in 68 conjugacy classes, 39 normal (27 characteristic)
C1, C2, C2 [×4], C3, C4 [×2], C4 [×2], C22, C22 [×4], S3 [×2], S3, C6, C6, C8 [×2], C8 [×2], C2×C4, C2×C4 [×5], C23, Dic3 [×2], C12 [×2], D6 [×2], D6 [×2], C2×C6, C2×C8 [×2], M4(2), M4(2) [×3], C22×C4, C3⋊C8 [×2], C24 [×2], C4×S3 [×4], C2×Dic3, C2×C12, C22×S3, C2×M4(2), S3×C8 [×2], C8⋊S3 [×2], C4.Dic3, C3×M4(2), S3×C2×C4, S3×M4(2)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], C23, D6 [×3], M4(2) [×2], C22×C4, C4×S3 [×2], C22×S3, C2×M4(2), S3×C2×C4, S3×M4(2)

Character table of S3×M4(2)

 class 12A2B2C2D2E34A4B4C4D4E4F6A6B8A8B8C8D8E8F8G8H12A12B12C24A24B24C24D
 size 112336211233624222266662244444
ρ1111111111111111111111111111111    trivial
ρ2111111111111111-1-1-1-1-1-1-1-1111-1-1-1-1    linear of order 2
ρ3111-1-1-11111-1-1-1111111-1-1-1-11111111    linear of order 2
ρ4111-1-1-11111-1-1-111-1-1-1-11111111-1-1-1-1    linear of order 2
ρ511-111-1111-111-11-1-1-1111-11-111-11-11-1    linear of order 2
ρ611-111-1111-111-11-111-1-1-11-1111-1-11-11    linear of order 2
ρ711-1-1-11111-1-1-111-1-1-111-11-1111-11-11-1    linear of order 2
ρ811-1-1-11111-1-1-111-111-1-11-11-111-1-11-11    linear of order 2
ρ911-111-11-1-11-1-111-1i-ii-i-iii-i-1-11-i-iii    linear of order 4
ρ101111111-1-1-1-1-1-111i-i-iiii-i-i-1-1-1i-i-ii    linear of order 4
ρ11111-1-1-11-1-1-111111-iii-iii-i-i-1-1-1-iii-i    linear of order 4
ρ1211-1-1-111-1-1111-11-1-ii-ii-iii-i-1-11ii-i-i    linear of order 4
ρ131111111-1-1-1-1-1-111-iii-i-i-iii-1-1-1-iii-i    linear of order 4
ρ1411-111-11-1-11-1-111-1-ii-iii-i-ii-1-11ii-i-i    linear of order 4
ρ1511-1-1-111-1-1111-11-1i-ii-ii-i-ii-1-11-i-iii    linear of order 4
ρ16111-1-1-11-1-1-111111i-i-ii-i-iii-1-1-1i-i-ii    linear of order 4
ρ1722-2000-122-2000-11-2-2220000-1-11-11-11    orthogonal lifted from D6
ρ18222000-1222000-1-1-2-2-2-20000-1-1-11111    orthogonal lifted from D6
ρ1922-2000-122-2000-1122-2-20000-1-111-11-1    orthogonal lifted from D6
ρ20222000-1222000-1-122220000-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ2122-2000-1-2-22000-112i-2i2i-2i000011-1ii-i-i    complex lifted from C4×S3
ρ222-202-202-2i2i02i-2i0-2000000000-2i2i00000    complex lifted from M4(2)
ρ232-20-22022i-2i02i-2i0-20000000002i-2i00000    complex lifted from M4(2)
ρ24222000-1-2-2-2000-1-12i-2i-2i2i0000111-iii-i    complex lifted from C4×S3
ρ2522-2000-1-2-22000-11-2i2i-2i2i000011-1-i-iii    complex lifted from C4×S3
ρ262-20-2202-2i2i0-2i2i0-2000000000-2i2i00000    complex lifted from M4(2)
ρ27222000-1-2-2-2000-1-1-2i2i2i-2i0000111i-i-ii    complex lifted from C4×S3
ρ282-202-2022i-2i0-2i2i0-20000000002i-2i00000    complex lifted from M4(2)
ρ294-40000-24i-4i00002000000000-2i2i00000    complex faithful
ρ304-40000-2-4i4i000020000000002i-2i00000    complex faithful

Permutation representations of S3×M4(2)
On 24 points - transitive group 24T104
Generators in S24
(1 13 22)(2 14 23)(3 15 24)(4 16 17)(5 9 18)(6 10 19)(7 11 20)(8 12 21)
(9 18)(10 19)(11 20)(12 21)(13 22)(14 23)(15 24)(16 17)
(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 6)(4 8)(10 14)(12 16)(17 21)(19 23)

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

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

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

G:=TransitiveGroup(24,104);

S3×M4(2) is a maximal subgroup of
M4(2).19D6  M4(2).21D6  C423D6  M4(2).25D6  M4(2)⋊26D6  M4(2)⋊28D6  D84D6  D24⋊C22  C24⋊D6  C3⋊C820D6  C40⋊D6  D154M4(2)  D15⋊M4(2)  D152M4(2)
S3×M4(2) is a maximal quotient of
C24⋊Q8  C42.182D6  C89D12  Dic35M4(2)  Dic3.5M4(2)  Dic3.M4(2)  D6⋊M4(2)  D62M4(2)  Dic3⋊M4(2)  C42.27D6  C42.200D6  C42.202D6  D63M4(2)  C12⋊M4(2)  Dic34M4(2)  D66M4(2)  C24⋊D4  C2421D4  C24⋊D6  C3⋊C820D6  C40⋊D6  D154M4(2)  D15⋊M4(2)  D152M4(2)

Matrix representation of S3×M4(2) in GL4(𝔽5) generated by

0002
0420
0200
2004
,
1000
0100
0340
2004
,
0300
1000
3001
0130
,
1000
0400
0040
0001
G:=sub<GL(4,GF(5))| [0,0,0,2,0,4,2,0,0,2,0,0,2,0,0,4],[1,0,0,2,0,1,3,0,0,0,4,0,0,0,0,4],[0,1,3,0,3,0,0,1,0,0,0,3,0,0,1,0],[1,0,0,0,0,4,0,0,0,0,4,0,0,0,0,1] >;

S3×M4(2) in GAP, Magma, Sage, TeX

S_3\times M_4(2)
% in TeX

G:=Group("S3xM4(2)");
// GroupNames label

G:=SmallGroup(96,113);
// by ID

G=gap.SmallGroup(96,113);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,188,50,69,2309]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^2=c^8=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^5>;
// generators/relations

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Character table of S3×M4(2) in TeX

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