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G = D4.5S4order 192 = 26·3

2nd non-split extension by D4 of S4 acting through Inn(D4)

non-abelian, soluble

Aliases: D4.5S4, 2- 1+4:3S3, GL2(F3).C22, CSU2(F3).C22, SL2(F3).9C23, (C2xQ8).D6, C4.15(C2xS4), D4.A4:3C2, C4oD4.7D6, C4.6S4:4C2, C4.S4:5C2, C22.6(C2xS4), C2.20(C22xS4), Q8.D6:2C2, C4.A4.6C22, Q8.10(C22xS3), (C2xCSU2(F3)):6C2, (C2xSL2(F3)).C22, SmallGroup(192,1486)

Series: Derived Chief Lower central Upper central

C1C2Q8SL2(F3) — D4.5S4
C1C2Q8SL2(F3)GL2(F3)C4.6S4 — D4.5S4
SL2(F3) — D4.5S4
C1C2D4

Generators and relations for D4.5S4
 G = < a,b,c,d,e,f | a4=b2=e3=f2=1, c2=d2=a2, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf=a2b, dcd-1=a2c, ece-1=a2cd, fcf=cd, ede-1=c, fdf=a2d, fef=e-1 >

Subgroups: 471 in 140 conjugacy classes, 27 normal (15 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C2xC4, D4, D4, Q8, Q8, Dic3, C12, D6, C2xC6, C2xC8, M4(2), D8, SD16, Q16, C2xQ8, C2xQ8, C4oD4, C4oD4, SL2(F3), Dic6, C4xS3, C2xDic3, C3:D4, C3xD4, C8oD4, C2xQ16, C4oD8, C8.C22, 2- 1+4, 2- 1+4, CSU2(F3), CSU2(F3), GL2(F3), C2xSL2(F3), C4.A4, D4:2S3, Q8oD8, C2xCSU2(F3), Q8.D6, C4.S4, C4.6S4, D4.A4, D4.5S4
Quotients: C1, C2, C22, S3, C23, D6, S4, C22xS3, C2xS4, C22xS4, D4.5S4

Character table of D4.5S4

 class 12A2B2C2D2E34A4B4C4D4E4F4G6A6B6C8A8B8C8D8E12
 size 112261282666121212816166612121216
ρ111111111111111111111111    trivial
ρ2111-1-111-1-1111-1-111-1-1-1-111-1    linear of order 2
ρ311-1-11111-11-1-1-111-1-111-11-11    linear of order 2
ρ411-11-111-111-1-11-11-11-1-111-1-1    linear of order 2
ρ5111-1-1-11-1-111-11111-1111-1-1-1    linear of order 2
ρ611111-111111-1-1-1111-1-1-1-1-11    linear of order 2
ρ711-11-1-11-111-11-111-1111-1-11-1    linear of order 2
ρ811-1-11-111-11-111-11-1-1-1-11-111    linear of order 2
ρ922-22-20-1-222-2000-11-1000001    orthogonal lifted from D6
ρ10222-2-20-1-2-222000-1-11000001    orthogonal lifted from D6
ρ1122-2-220-12-22-2000-11100000-1    orthogonal lifted from D6
ρ12222220-12222000-1-1-100000-1    orthogonal lifted from S3
ρ1333-33110-3-1-11-11-100011-1-110    orthogonal lifted from C2xS4
ρ1433-3-3-11031-11-1-11000-1-11-110    orthogonal lifted from C2xS4
ρ15333-31-10-31-1-1-111000-1-1-1110    orthogonal lifted from C2xS4
ρ163333-1-103-1-1-1-1-1-1000111110    orthogonal lifted from S4
ρ1733-331-10-3-1-111-11000-1-111-10    orthogonal lifted from C2xS4
ρ1833-3-3-1-1031-1111-100011-11-10    orthogonal lifted from C2xS4
ρ19333-3110-31-1-11-1-1000111-1-10    orthogonal lifted from C2xS4
ρ203333-1103-1-1-1111000-1-1-1-1-10    orthogonal lifted from S4
ρ214-40000-20000000200-22220000    symplectic faithful, Schur index 2
ρ224-40000-2000000020022-220000    symplectic faithful, Schur index 2
ρ238-8000020000000-200000000    symplectic faithful, Schur index 2

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

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

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

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

Matrix representation of D4.5S4 in GL4(F7) generated by

6205
5043
2543
6644
,
4052
5003
2543
5516
,
6611
2041
3301
4351
,
4065
6011
5524
6141
,
1141
0333
6602
1631
,
6636
6344
6363
5046
G:=sub<GL(4,GF(7))| [6,5,2,6,2,0,5,6,0,4,4,4,5,3,3,4],[4,5,2,5,0,0,5,5,5,0,4,1,2,3,3,6],[6,2,3,4,6,0,3,3,1,4,0,5,1,1,1,1],[4,6,5,6,0,0,5,1,6,1,2,4,5,1,4,1],[1,0,6,1,1,3,6,6,4,3,0,3,1,3,2,1],[6,6,6,5,6,3,3,0,3,4,6,4,6,4,3,6] >;

D4.5S4 in GAP, Magma, Sage, TeX

D_4._5S_4
% in TeX

G:=Group("D4.5S4");
// GroupNames label

G:=SmallGroup(192,1486);
// by ID

G=gap.SmallGroup(192,1486);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,2,-2,680,2102,1059,451,1684,655,172,1013,404,285,124]);
// 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=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,f*b*f=a^2*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 D4.5S4 in TeX

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