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G = S3xM4(2)  order 96 = 25·3

Direct product of S3 and M4(2)

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

Aliases: S3xM4(2), C8:6D6, C24:7C22, Dic3oM4(2), C12.38C23, (S3xC8):7C2, C8:S3:5C2, (C4xS3).1C4, C4.15(C4xS3), C3:C8:11C22, D6.6(C2xC4), (C2xC4).45D6, C3:2(C2xM4(2)), C12.12(C2xC4), C4.Dic3:5C2, C22.7(C4xS3), (C3xM4(2)):5C2, (C22xS3).4C4, C6.15(C22xC4), C4.38(C22xS3), (C2xDic3).6C4, Dic3.7(C2xC4), (C4xS3).18C22, (C2xC12).25C22, (S3xC2xC4).4C2, C2.16(S3xC2xC4), (C2xC6).5(C2xC4), SmallGroup(96,113)

Series: Derived Chief Lower central Upper central

C1C6 — S3xM4(2)
C1C3C6C12C4xS3S3xC2xC4 — S3xM4(2)
C3C6 — S3xM4(2)
C1C4M4(2)

Generators and relations for S3xM4(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, C3, C4, C4, C22, C22, S3, S3, C6, C6, C8, C8, C2xC4, C2xC4, C23, Dic3, C12, D6, D6, C2xC6, C2xC8, M4(2), M4(2), C22xC4, C3:C8, C24, C4xS3, C2xDic3, C2xC12, C22xS3, C2xM4(2), S3xC8, C8:S3, C4.Dic3, C3xM4(2), S3xC2xC4, S3xM4(2)
Quotients: C1, C2, C4, C22, S3, C2xC4, C23, D6, M4(2), C22xC4, C4xS3, C22xS3, C2xM4(2), S3xC2xC4, S3xM4(2)

Character table of S3xM4(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 C4xS3
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 C4xS3
ρ2522-2000-1-2-22000-11-2i2i-2i2i000011-1-i-iii    complex lifted from C4xS3
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 C4xS3
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 S3xM4(2)
On 24 points - transitive group 24T104
Generators in S24
(1 9 22)(2 10 23)(3 11 24)(4 12 17)(5 13 18)(6 14 19)(7 15 20)(8 16 21)
(9 22)(10 23)(11 24)(12 17)(13 18)(14 19)(15 20)(16 21)
(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)| (1,9,22)(2,10,23)(3,11,24)(4,12,17)(5,13,18)(6,14,19)(7,15,20)(8,16,21), (9,22)(10,23)(11,24)(12,17)(13,18)(14,19)(15,20)(16,21), (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( (1,9,22)(2,10,23)(3,11,24)(4,12,17)(5,13,18)(6,14,19)(7,15,20)(8,16,21), (9,22)(10,23)(11,24)(12,17)(13,18)(14,19)(15,20)(16,21), (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([[(1,9,22),(2,10,23),(3,11,24),(4,12,17),(5,13,18),(6,14,19),(7,15,20),(8,16,21)], [(9,22),(10,23),(11,24),(12,17),(13,18),(14,19),(15,20),(16,21)], [(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);

S3xM4(2) is a maximal subgroup of
M4(2).19D6  M4(2).21D6  C42:3D6  M4(2).25D6  M4(2):26D6  M4(2):28D6  D8:4D6  D24:C22  C24:D6  C3:C8:20D6  C40:D6  D15:4M4(2)  D15:M4(2)  D15:2M4(2)
S3xM4(2) is a maximal quotient of
C24:Q8  C42.182D6  C8:9D12  Dic3:5M4(2)  Dic3.5M4(2)  Dic3.M4(2)  D6:M4(2)  D6:2M4(2)  Dic3:M4(2)  C42.27D6  C42.200D6  C42.202D6  D6:3M4(2)  C12:M4(2)  Dic3:4M4(2)  D6:6M4(2)  C24:D4  C24:21D4  C24:D6  C3:C8:20D6  C40:D6  D15:4M4(2)  D15:M4(2)  D15:2M4(2)

Matrix representation of S3xM4(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] >;

S3xM4(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

Export

Character table of S3xM4(2) in TeX

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