Copied to
clipboard

G = D4.3S4  order 192 = 26·3

3rd non-split extension by D4 of S4 acting via S4/A4=C2

non-abelian, soluble

Aliases: D4.3S4, U2(𝔽3)⋊4C2, 2- 1+4⋊1S3, SL2(𝔽3).10D4, C4.8(C2×S4), D4.A4⋊1C2, C4○D4.4D6, C4.3S4⋊3C2, Q8.7(C3⋊D4), C4.A4.4C22, C2.15(A4⋊D4), SmallGroup(192,990)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2 — Q8 — C4.A4 — D4.3S4
Chief series
C1 — C2 — Q8 — SL2(𝔽3) — C4.A4 — C4.3S4 — D4.3S4
Lower central
SL2(𝔽3) — C4.A4 — D4.3S4
Upper central
C1 — C2 — C4 — D4

Generators and relations for D4.3S4
 G = < a,b,c,d,e,f | a4=b2=e3=f2=1, c2=d2=a2, bab=faf=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, fbf=ab, dcd-1=a2c, ece-1=a2cd, fcf=cd, ede-1=c, fdf=a2d, fef=e-1 >

Subgroups: 341 in 72 conjugacy classes, 13 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C8, C2×C4, D4, D4, Q8, Q8, C23, C12, D6, C2×C6, C42, C22⋊C4, M4(2), D8, SD16, C2×D4, C2×Q8, C4○D4, C4○D4, C3⋊C8, SL2(𝔽3), D12, C3×D4, C4.10D4, C4≀C2, C4.4D4, C8⋊C22, 2- 1+4, D4⋊S3, GL2(𝔽3), C2×SL2(𝔽3), C4.A4, D4.8D4, U2(𝔽3), C4.3S4, D4.A4, D4.3S4
Quotients: C1, C2, C22, S3, D4, D6, C3⋊D4, S4, C2×S4, A4⋊D4, D4.3S4

Character table of D4.3S4

 class 12A2B2C2D34A4B4C4D4E6A6B6C8A8B12
 size 11462482612121281616242416
ρ111111111111111111    trivial
ρ211-11-111111-11-1-11-11    linear of order 2
ρ31111-1111-1-11111-1-11    linear of order 2
ρ411-111111-1-1-11-1-1-111    linear of order 2
ρ522-220-12200-2-11100-1    orthogonal lifted from D6
ρ622220-122002-1-1-100-1    orthogonal lifted from S3
ρ7220-202-2200020000-2    orthogonal lifted from D4
ρ8220-20-1-22000-1√-3-√-3001    complex lifted from C3⋊D4
ρ9220-20-1-22000-1-√-3√-3001    complex lifted from C3⋊D4
ρ10333-1103-1-1-1-10001-10    orthogonal lifted from S4
ρ1133-3-1-103-1-1-11000110    orthogonal lifted from C2×S4
ρ1233-3-1103-1111000-1-10    orthogonal lifted from C2×S4
ρ13333-1-103-111-1000-110    orthogonal lifted from S4
ρ144-4000-2002i-2i0200000    complex faithful
ρ154-4000-200-2i2i0200000    complex faithful
ρ16660200-6-2000000000    orthogonal lifted from A4⋊D4
ρ178-8000200000-200000    orthogonal faithful, Schur index 2

Smallest permutation representation of D4.3S4
►On 32 points
Generators in S32
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)(25 26 27 28)(29 30 31 32)

(1 21)(2 24)(3 23)(4 22)(5 28)(6 27)(7 26)(8 25)(9 19)(10 18)(11 17)(12 20)(13 30)(14 29)(15 32)(16 31)

(1 8 3 6)(2 5 4 7)(9 31 11 29)(10 32 12 30)(13 18 15 20)(14 19 16 17)(21 25 23 27)(22 26 24 28)

(1 18 3 20)(2 19 4 17)(5 14 7 16)(6 15 8 13)(9 22 11 24)(10 23 12 21)(25 30 27 32)(26 31 28 29)

(5 19 16)(6 20 13)(7 17 14)(8 18 15)(9 31 28)(10 32 25)(11 29 26)(12 30 27)

(2 4)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(18 20)(21 24)(22 23)(25 29)(26 32)(27 31)(28 30)
 
G:=sub<Sym(32)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32), (1,21)(2,24)(3,23)(4,22)(5,28)(6,27)(7,26)(8,25)(9,19)(10,18)(11,17)(12,20)(13,30)(14,29)(15,32)(16,31), (1,8,3,6)(2,5,4,7)(9,31,11,29)(10,32,12,30)(13,18,15,20)(14,19,16,17)(21,25,23,27)(22,26,24,28), (1,18,3,20)(2,19,4,17)(5,14,7,16)(6,15,8,13)(9,22,11,24)(10,23,12,21)(25,30,27,32)(26,31,28,29), (5,19,16)(6,20,13)(7,17,14)(8,18,15)(9,31,28)(10,32,25)(11,29,26)(12,30,27), (2,4)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(18,20)(21,24)(22,23)(25,29)(26,32)(27,31)(28,30)>;
 
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)(25,26,27,28)(29,30,31,32), (1,21)(2,24)(3,23)(4,22)(5,28)(6,27)(7,26)(8,25)(9,19)(10,18)(11,17)(12,20)(13,30)(14,29)(15,32)(16,31), (1,8,3,6)(2,5,4,7)(9,31,11,29)(10,32,12,30)(13,18,15,20)(14,19,16,17)(21,25,23,27)(22,26,24,28), (1,18,3,20)(2,19,4,17)(5,14,7,16)(6,15,8,13)(9,22,11,24)(10,23,12,21)(25,30,27,32)(26,31,28,29), (5,19,16)(6,20,13)(7,17,14)(8,18,15)(9,31,28)(10,32,25)(11,29,26)(12,30,27), (2,4)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(18,20)(21,24)(22,23)(25,29)(26,32)(27,31)(28,30) );
 
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),(25,26,27,28),(29,30,31,32)], [(1,21),(2,24),(3,23),(4,22),(5,28),(6,27),(7,26),(8,25),(9,19),(10,18),(11,17),(12,20),(13,30),(14,29),(15,32),(16,31)], [(1,8,3,6),(2,5,4,7),(9,31,11,29),(10,32,12,30),(13,18,15,20),(14,19,16,17),(21,25,23,27),(22,26,24,28)], [(1,18,3,20),(2,19,4,17),(5,14,7,16),(6,15,8,13),(9,22,11,24),(10,23,12,21),(25,30,27,32),(26,31,28,29)], [(5,19,16),(6,20,13),(7,17,14),(8,18,15),(9,31,28),(10,32,25),(11,29,26),(12,30,27)], [(2,4),(5,16),(6,15),(7,14),(8,13),(9,12),(10,11),(18,20),(21,24),(22,23),(25,29),(26,32),(27,31),(28,30)]])
 

Matrix representation of D4.3S4 ►in GL4(𝔽5) generated by

2000
0300
0020
0003
,
0304
3010
0203
3030
,
3000
0102
0020
0404
,
0040
0200
1000
0303
,
3020
0401
1010
0400
,
0004
1030
0202
4000
G:=sub<GL(4,GF(5))| [2,0,0,0,0,3,0,0,0,0,2,0,0,0,0,3],[0,3,0,3,3,0,2,0,0,1,0,3,4,0,3,0],[3,0,0,0,0,1,0,4,0,0,2,0,0,2,0,4],[0,0,1,0,0,2,0,3,4,0,0,0,0,0,0,3],[3,0,1,0,0,4,0,4,2,0,1,0,0,1,0,0],[0,1,0,4,0,0,2,0,0,3,0,0,4,0,2,0] >;
 

D4.3S4 in GAP, Magma, Sage, TeX 

D_4._3S_4
 
% in TeX
 
G:=Group("D4.3S4");
 
// GroupNames label
 
G:=SmallGroup(192,990);
 
// by ID
 
G=gap.SmallGroup(192,990);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-3,-2,2,-2,85,2102,1059,520,451,1684,655,172,1013,404,285,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^4=b^2=e^3=f^2=1,c^2=d^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^-1=a^2*c,e*c*e^-1=a^2*c*d,f*c*f=c*d,e*d*e^-1=c,f*d*f=a^2*d,f*e*f=e^-1>;
 
// generators/relations
 

Export

Character table of D4.3S4 in TeX

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁