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G = C8.4S4order 192 = 26·3

4th non-split extension by C8 of S4 acting via S4/A4=C2

non-abelian, soluble

Aliases: C8.4S4, Q8.4D12, SL2(F3).4D4, C8oD4:1S3, C8.A4:2C2, C4.20(C2xS4), C4.3S4.C2, C2.10(C4:S4), C4.S4:1C2, C4oD4.11D6, C4.A4.8C22, SmallGroup(192,965)

Series: Derived Chief Lower central Upper central

C1C2Q8C4.A4 — C8.4S4
C1C2Q8SL2(F3)C4.A4C4.3S4 — C8.4S4
SL2(F3)C4.A4 — C8.4S4
C1C2C4C8

Generators and relations for C8.4S4
 G = < a,b,c,d,e | a8=d3=e2=1, b2=c2=a4, ab=ba, ac=ca, ad=da, eae=a3, cbc-1=a4b, dbd-1=a4bc, ebe=bc, dcd-1=b, ece=a4c, ede=d-1 >

Subgroups: 293 in 62 conjugacy classes, 13 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C8, C8, C2xC4, D4, Q8, Q8, C23, Dic3, C12, D6, C2xC8, M4(2), D8, SD16, Q16, C2xD4, C2xQ8, C4oD4, C24, SL2(F3), Dic6, D12, C4.D4, C4.10D4, C8.C4, C8oD4, C2xSD16, C8:C22, C8.C22, C24:C2, CSU2(F3), GL2(F3), C4.A4, D4.3D4, C8.A4, C4.S4, C4.3S4, C8.4S4
Quotients: C1, C2, C22, S3, D4, D6, D12, S4, C2xS4, C4:S4, C8.4S4

Character table of C8.4S4

 class 12A2B2C34A4B4C68A8B8C8D8E12A12B24A24B24C24D
 size 1162482624822122424888888
ρ111111111111111111111    trivial
ρ2111-111111-1-1-11-111-1-1-1-1    linear of order 2
ρ3111-1111-11111-1-1111111    linear of order 2
ρ41111111-11-1-1-1-1111-1-1-1-1    linear of order 2
ρ522-202-220200000-2-20000    orthogonal lifted from D4
ρ62220-1220-1-2-2-200-1-11111    orthogonal lifted from D6
ρ72220-1220-122200-1-1-1-1-1-1    orthogonal lifted from S3
ρ822-20-1-220-100000113-33-3    orthogonal lifted from D12
ρ922-20-1-220-10000011-33-33    orthogonal lifted from D12
ρ1033-1103-11033-1-1-1000000    orthogonal lifted from S4
ρ1133-1-103-1-1033-111000000    orthogonal lifted from S4
ρ1233-1103-1-10-3-311-1000000    orthogonal lifted from C2xS4
ρ1333-1-103-110-3-31-11000000    orthogonal lifted from C2xS4
ρ144-400-200022-2-2-200000-2-2--2--2    complex faithful
ρ154-400-20002-2-22-200000--2--2-2-2    complex faithful
ρ164-4001000-1-2-22-20003-3ζ83ζ3838ζ3ζ83ζ32838ζ32ζ87ζ38785ζ3ζ87ζ328785ζ32    complex faithful
ρ174-4001000-12-2-2-20003-3ζ87ζ38785ζ3ζ87ζ328785ζ32ζ83ζ3838ζ3ζ83ζ32838ζ32    complex faithful
ρ184-4001000-1-2-22-2000-33ζ83ζ32838ζ32ζ83ζ3838ζ3ζ87ζ328785ζ32ζ87ζ38785ζ3    complex faithful
ρ194-4001000-12-2-2-2000-33ζ87ζ328785ζ32ζ87ζ38785ζ3ζ83ζ32838ζ32ζ83ζ3838ζ3    complex faithful
ρ2066200-6-20000000000000    orthogonal lifted from C4:S4

Smallest permutation representation of C8.4S4
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 14 5 10)(2 15 6 11)(3 16 7 12)(4 9 8 13)(17 27 21 31)(18 28 22 32)(19 29 23 25)(20 30 24 26)
(1 25 5 29)(2 26 6 30)(3 27 7 31)(4 28 8 32)(9 18 13 22)(10 19 14 23)(11 20 15 24)(12 21 16 17)
(9 28 22)(10 29 23)(11 30 24)(12 31 17)(13 32 18)(14 25 19)(15 26 20)(16 27 21)
(2 4)(3 7)(6 8)(9 24)(10 19)(11 22)(12 17)(13 20)(14 23)(15 18)(16 21)(25 29)(26 32)(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), (1,14,5,10)(2,15,6,11)(3,16,7,12)(4,9,8,13)(17,27,21,31)(18,28,22,32)(19,29,23,25)(20,30,24,26), (1,25,5,29)(2,26,6,30)(3,27,7,31)(4,28,8,32)(9,18,13,22)(10,19,14,23)(11,20,15,24)(12,21,16,17), (9,28,22)(10,29,23)(11,30,24)(12,31,17)(13,32,18)(14,25,19)(15,26,20)(16,27,21), (2,4)(3,7)(6,8)(9,24)(10,19)(11,22)(12,17)(13,20)(14,23)(15,18)(16,21)(25,29)(26,32)(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), (1,14,5,10)(2,15,6,11)(3,16,7,12)(4,9,8,13)(17,27,21,31)(18,28,22,32)(19,29,23,25)(20,30,24,26), (1,25,5,29)(2,26,6,30)(3,27,7,31)(4,28,8,32)(9,18,13,22)(10,19,14,23)(11,20,15,24)(12,21,16,17), (9,28,22)(10,29,23)(11,30,24)(12,31,17)(13,32,18)(14,25,19)(15,26,20)(16,27,21), (2,4)(3,7)(6,8)(9,24)(10,19)(11,22)(12,17)(13,20)(14,23)(15,18)(16,21)(25,29)(26,32)(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)], [(1,14,5,10),(2,15,6,11),(3,16,7,12),(4,9,8,13),(17,27,21,31),(18,28,22,32),(19,29,23,25),(20,30,24,26)], [(1,25,5,29),(2,26,6,30),(3,27,7,31),(4,28,8,32),(9,18,13,22),(10,19,14,23),(11,20,15,24),(12,21,16,17)], [(9,28,22),(10,29,23),(11,30,24),(12,31,17),(13,32,18),(14,25,19),(15,26,20),(16,27,21)], [(2,4),(3,7),(6,8),(9,24),(10,19),(11,22),(12,17),(13,20),(14,23),(15,18),(16,21),(25,29),(26,32),(28,30)]])

Matrix representation of C8.4S4 in GL4(F3) generated by

0122
0001
2021
0102
,
0112
0120
0220
1010
,
2221
0102
2212
0202
,
2112
1212
0101
2010
,
1000
0100
1020
0202
G:=sub<GL(4,GF(3))| [0,0,2,0,1,0,0,1,2,0,2,0,2,1,1,2],[0,0,0,1,1,1,2,0,1,2,2,1,2,0,0,0],[2,0,2,0,2,1,2,2,2,0,1,0,1,2,2,2],[2,1,0,2,1,2,1,0,1,1,0,1,2,2,1,0],[1,0,1,0,0,1,0,2,0,0,2,0,0,0,0,2] >;

C8.4S4 in GAP, Magma, Sage, TeX

C_8._4S_4
% in TeX

G:=Group("C8.4S4");
// GroupNames label

G:=SmallGroup(192,965);
// by ID

G=gap.SmallGroup(192,965);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,2,-2,85,36,2102,520,451,1684,655,172,1013,404,285,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^8=d^3=e^2=1,b^2=c^2=a^4,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e=a^3,c*b*c^-1=a^4*b,d*b*d^-1=a^4*b*c,e*b*e=b*c,d*c*d^-1=b,e*c*e=a^4*c,e*d*e=d^-1>;
// generators/relations

Export

Character table of C8.4S4 in TeX

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