Aliases: C8.5S4, U2(𝔽3)⋊5C2, GL2(𝔽3)⋊2C4, CSU2(𝔽3)⋊2C4, C8.A4⋊6C2, C8○D4⋊4S3, C2.9(C4×S4), C4.29(C2×S4), C4.6S4.C2, Q8.5(C4×S3), C4○D4.10D6, C4.A4.12C22, SL2(𝔽3).4(C2×C4), SmallGroup(192,964)
Series: Derived ►Chief ►Lower central ►Upper central
| SL2(𝔽3) — C8.5S4 |
Generators and relations for C8.5S4
G = < a,b,c,d,e | a8=d3=e2=1, b2=c2=a4, ab=ba, ac=ca, ad=da, eae=a5, cbc-1=a4b, dbd-1=a4bc, ebe=bc, dcd-1=b, ece=a4c, ede=d-1 >
Subgroups: 215 in 62 conjugacy classes, 15 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C8, C8, C2×C4, D4, Q8, Q8, Dic3, C12, D6, C42, C2×C8, M4(2), D8, SD16, Q16, C4○D4, C4○D4, C3⋊C8, C24, SL2(𝔽3), C4×S3, C8⋊C4, C4≀C2, C8.C4, C8○D4, C8○D4, C4○D8, C8⋊S3, CSU2(𝔽3), GL2(𝔽3), C4.A4, C8.26D4, U2(𝔽3), C8.A4, C4.6S4, C8.5S4
Quotients: C1, C2, C4, C22, S3, C2×C4, D6, C4×S3, S4, C2×S4, C4×S4, C8.5S4
Character table of C8.5S4
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 6 | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | 12B | 24A | 24B | 24C | 24D | |
| size | 1 | 1 | 6 | 12 | 8 | 1 | 1 | 6 | 12 | 12 | 12 | 8 | 2 | 2 | 6 | 6 | 12 | 12 | 12 | 12 | 8 | 8 | 8 | 8 | 8 | 8 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ4 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ5 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | i | -i | -1 | 1 | -i | i | -i | i | i | -1 | -i | 1 | -1 | -1 | -i | i | -i | i | linear of order 4 |
| ρ6 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | -i | i | -1 | 1 | i | -i | i | -i | -i | -1 | i | 1 | -1 | -1 | i | -i | i | -i | linear of order 4 |
| ρ7 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | -i | i | 1 | 1 | -i | i | -i | i | -i | 1 | i | -1 | -1 | -1 | -i | i | -i | i | linear of order 4 |
| ρ8 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | i | -i | 1 | 1 | i | -i | i | -i | i | 1 | -i | -1 | -1 | -1 | i | -i | i | -i | linear of order 4 |
| ρ9 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | 2 | 0 | 0 | 0 | -1 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ10 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | 2 | 0 | 0 | 0 | -1 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ11 | 2 | 2 | -2 | 0 | -1 | -2 | -2 | 2 | 0 | 0 | 0 | -1 | 2i | -2i | 2i | -2i | 0 | 0 | 0 | 0 | 1 | 1 | -i | i | -i | i | complex lifted from C4×S3 |
| ρ12 | 2 | 2 | -2 | 0 | -1 | -2 | -2 | 2 | 0 | 0 | 0 | -1 | -2i | 2i | -2i | 2i | 0 | 0 | 0 | 0 | 1 | 1 | i | -i | i | -i | complex lifted from C4×S3 |
| ρ13 | 3 | 3 | -1 | 1 | 0 | 3 | 3 | -1 | 1 | 1 | 1 | 0 | -3 | -3 | 1 | 1 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ14 | 3 | 3 | -1 | -1 | 0 | 3 | 3 | -1 | -1 | -1 | -1 | 0 | -3 | -3 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ15 | 3 | 3 | -1 | 1 | 0 | 3 | 3 | -1 | -1 | -1 | 1 | 0 | 3 | 3 | -1 | -1 | 1 | -1 | 1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ16 | 3 | 3 | -1 | -1 | 0 | 3 | 3 | -1 | 1 | 1 | -1 | 0 | 3 | 3 | -1 | -1 | -1 | 1 | -1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ17 | 3 | 3 | 1 | -1 | 0 | -3 | -3 | -1 | -i | i | 1 | 0 | 3i | -3i | -i | i | i | -1 | -i | 1 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4×S4 |
| ρ18 | 3 | 3 | 1 | 1 | 0 | -3 | -3 | -1 | i | -i | -1 | 0 | 3i | -3i | -i | i | -i | 1 | i | -1 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4×S4 |
| ρ19 | 3 | 3 | 1 | -1 | 0 | -3 | -3 | -1 | i | -i | 1 | 0 | -3i | 3i | i | -i | -i | -1 | i | 1 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4×S4 |
| ρ20 | 3 | 3 | 1 | 1 | 0 | -3 | -3 | -1 | -i | i | -1 | 0 | -3i | 3i | i | -i | i | 1 | -i | -1 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4×S4 |
| ρ21 | 4 | -4 | 0 | 0 | -2 | -4i | 4i | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | complex faithful |
| ρ22 | 4 | -4 | 0 | 0 | -2 | 4i | -4i | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | complex faithful |
| ρ23 | 4 | -4 | 0 | 0 | 1 | -4i | 4i | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | i | -i | 2ζ8ζ3+ζ8 | 2ζ87ζ3+ζ87 | 2ζ85ζ3+ζ85 | 2ζ83ζ3+ζ83 | complex faithful |
| ρ24 | 4 | -4 | 0 | 0 | 1 | 4i | -4i | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -i | i | 2ζ87ζ3+ζ87 | 2ζ8ζ3+ζ8 | 2ζ83ζ3+ζ83 | 2ζ85ζ3+ζ85 | complex faithful |
| ρ25 | 4 | -4 | 0 | 0 | 1 | 4i | -4i | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -i | i | 2ζ83ζ3+ζ83 | 2ζ85ζ3+ζ85 | 2ζ87ζ3+ζ87 | 2ζ8ζ3+ζ8 | complex faithful |
| ρ26 | 4 | -4 | 0 | 0 | 1 | -4i | 4i | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | i | -i | 2ζ85ζ3+ζ85 | 2ζ83ζ3+ζ83 | 2ζ8ζ3+ζ8 | 2ζ87ζ3+ζ87 | complex faithful |
►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 6)(4 8)(9 22)(10 19)(11 24)(12 21)(13 18)(14 23)(15 20)(16 17)(25 29)(27 31)
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), (2,6)(4,8)(9,22)(10,19)(11,24)(12,21)(13,18)(14,23)(15,20)(16,17)(25,29)(27,31)>;
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), (2,6)(4,8)(9,22)(10,19)(11,24)(12,21)(13,18)(14,23)(15,20)(16,17)(25,29)(27,31) );
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)], [(2,6),(4,8),(9,22),(10,19),(11,24),(12,21),(13,18),(14,23),(15,20),(16,17),(25,29),(27,31)]])
Matrix representation of C8.5S4 ►in GL4(𝔽5) generated by
| 0 | 0 | 0 | 4 |
| 0 | 0 | 4 | 3 |
| 4 | 3 | 0 | 0 |
| 3 | 0 | 0 | 0 |
| 1 | 3 | 0 | 0 |
| 1 | 4 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 3 | 2 |
| 2 | 1 | 0 | 0 |
| 0 | 3 | 0 | 0 |
| 0 | 0 | 1 | 3 |
| 0 | 0 | 1 | 4 |
| 2 | 3 | 0 | 0 |
| 1 | 2 | 0 | 0 |
| 0 | 0 | 1 | 4 |
| 0 | 0 | 3 | 3 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 2 | 2 |
| 1 | 0 | 0 | 0 |
| 4 | 3 | 0 | 0 |
G:=sub<GL(4,GF(5))| [0,0,4,3,0,0,3,0,0,4,0,0,4,3,0,0],[1,1,0,0,3,4,0,0,0,0,3,3,0,0,0,2],[2,0,0,0,1,3,0,0,0,0,1,1,0,0,3,4],[2,1,0,0,3,2,0,0,0,0,1,3,0,0,4,3],[0,0,1,4,0,0,0,3,1,2,0,0,0,2,0,0] >;
C8.5S4 in GAP, Magma, Sage, TeX
C_8._5S_4
% in TeX
G:=Group("C8.5S4"); // GroupNames label
G:=SmallGroup(192,964);
// by ID
G=gap.SmallGroup(192,964);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,2,-2,1373,36,520,451,1684,655,172,1013,404,285,124]);
// 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^5,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
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