metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C51⋊1C8, D17.Dic3, C17⋊(C3⋊C8), C3⋊(C17⋊C8), C17⋊C4.S3, (C3×D17).1C4, (C3×C17⋊C4).1C2, SmallGroup(408,34)
Series: Derived ►Chief ►Lower central ►Upper central
| C51 — C51⋊C8 |
Generators and relations for C51⋊C8
G = < a,b | a51=b8=1, bab-1=a26 >
Character table of C51⋊C8
| class | 1 | 2 | 3 | 4A | 4B | 6 | 8A | 8B | 8C | 8D | 12A | 12B | 17A | 17B | 51A | 51B | 51C | 51D | |
| size | 1 | 17 | 2 | 17 | 17 | 34 | 51 | 51 | 51 | 51 | 34 | 34 | 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 | trivial |
| ρ2 | 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 | i | -i | -i | i | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ4 | 1 | 1 | 1 | -1 | -1 | 1 | -i | i | i | -i | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ5 | 1 | -1 | 1 | i | -i | -1 | ζ83 | ζ8 | ζ85 | ζ87 | i | -i | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 8 |
| ρ6 | 1 | -1 | 1 | -i | i | -1 | ζ8 | ζ83 | ζ87 | ζ85 | -i | i | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 8 |
| ρ7 | 1 | -1 | 1 | i | -i | -1 | ζ87 | ζ85 | ζ8 | ζ83 | i | -i | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 8 |
| ρ8 | 1 | -1 | 1 | -i | i | -1 | ζ85 | ζ87 | ζ83 | ζ8 | -i | i | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 8 |
| ρ9 | 2 | 2 | -1 | 2 | 2 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | 2 | 2 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ10 | 2 | 2 | -1 | -2 | -2 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | 2 | 2 | -1 | -1 | -1 | -1 | symplectic lifted from Dic3, Schur index 2 |
| ρ11 | 2 | -2 | -1 | -2i | 2i | 1 | 0 | 0 | 0 | 0 | i | -i | 2 | 2 | -1 | -1 | -1 | -1 | complex lifted from C3⋊C8 |
| ρ12 | 2 | -2 | -1 | 2i | -2i | 1 | 0 | 0 | 0 | 0 | -i | i | 2 | 2 | -1 | -1 | -1 | -1 | complex lifted from C3⋊C8 |
| ρ13 | 8 | 0 | 8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√17/2 | -1+√17/2 | -1-√17/2 | -1+√17/2 | -1-√17/2 | -1+√17/2 | orthogonal lifted from C17⋊C8 |
| ρ14 | 8 | 0 | 8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√17/2 | -1-√17/2 | -1+√17/2 | -1-√17/2 | -1+√17/2 | -1-√17/2 | orthogonal lifted from C17⋊C8 |
| ρ15 | 8 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√17/2 | -1+√17/2 | -ζ3ζ1714-ζ3ζ1712+ζ3ζ1711+ζ3ζ1710+ζ3ζ177+ζ3ζ176-ζ3ζ175-ζ3ζ173-ζ1714-ζ1712-ζ175-ζ173 | ζ32ζ1716-ζ32ζ1715+ζ32ζ1713-ζ32ζ179-ζ32ζ178+ζ32ζ174-ζ32ζ172+ζ32ζ17-ζ1715-ζ179-ζ178-ζ172 | ζ3ζ1714+ζ3ζ1712-ζ3ζ1711-ζ3ζ1710-ζ3ζ177-ζ3ζ176+ζ3ζ175+ζ3ζ173-ζ1711-ζ1710-ζ177-ζ176 | -ζ32ζ1716+ζ32ζ1715-ζ32ζ1713+ζ32ζ179+ζ32ζ178-ζ32ζ174+ζ32ζ172-ζ32ζ17-ζ1716-ζ1713-ζ174-ζ17 | complex faithful |
| ρ16 | 8 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√17/2 | -1-√17/2 | -ζ32ζ1716+ζ32ζ1715-ζ32ζ1713+ζ32ζ179+ζ32ζ178-ζ32ζ174+ζ32ζ172-ζ32ζ17-ζ1716-ζ1713-ζ174-ζ17 | -ζ3ζ1714-ζ3ζ1712+ζ3ζ1711+ζ3ζ1710+ζ3ζ177+ζ3ζ176-ζ3ζ175-ζ3ζ173-ζ1714-ζ1712-ζ175-ζ173 | ζ32ζ1716-ζ32ζ1715+ζ32ζ1713-ζ32ζ179-ζ32ζ178+ζ32ζ174-ζ32ζ172+ζ32ζ17-ζ1715-ζ179-ζ178-ζ172 | ζ3ζ1714+ζ3ζ1712-ζ3ζ1711-ζ3ζ1710-ζ3ζ177-ζ3ζ176+ζ3ζ175+ζ3ζ173-ζ1711-ζ1710-ζ177-ζ176 | complex faithful |
| ρ17 | 8 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√17/2 | -1-√17/2 | ζ32ζ1716-ζ32ζ1715+ζ32ζ1713-ζ32ζ179-ζ32ζ178+ζ32ζ174-ζ32ζ172+ζ32ζ17-ζ1715-ζ179-ζ178-ζ172 | ζ3ζ1714+ζ3ζ1712-ζ3ζ1711-ζ3ζ1710-ζ3ζ177-ζ3ζ176+ζ3ζ175+ζ3ζ173-ζ1711-ζ1710-ζ177-ζ176 | -ζ32ζ1716+ζ32ζ1715-ζ32ζ1713+ζ32ζ179+ζ32ζ178-ζ32ζ174+ζ32ζ172-ζ32ζ17-ζ1716-ζ1713-ζ174-ζ17 | -ζ3ζ1714-ζ3ζ1712+ζ3ζ1711+ζ3ζ1710+ζ3ζ177+ζ3ζ176-ζ3ζ175-ζ3ζ173-ζ1714-ζ1712-ζ175-ζ173 | complex faithful |
| ρ18 | 8 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√17/2 | -1+√17/2 | ζ3ζ1714+ζ3ζ1712-ζ3ζ1711-ζ3ζ1710-ζ3ζ177-ζ3ζ176+ζ3ζ175+ζ3ζ173-ζ1711-ζ1710-ζ177-ζ176 | -ζ32ζ1716+ζ32ζ1715-ζ32ζ1713+ζ32ζ179+ζ32ζ178-ζ32ζ174+ζ32ζ172-ζ32ζ17-ζ1716-ζ1713-ζ174-ζ17 | -ζ3ζ1714-ζ3ζ1712+ζ3ζ1711+ζ3ζ1710+ζ3ζ177+ζ3ζ176-ζ3ζ175-ζ3ζ173-ζ1714-ζ1712-ζ175-ζ173 | ζ32ζ1716-ζ32ζ1715+ζ32ζ1713-ζ32ζ179-ζ32ζ178+ζ32ζ174-ζ32ζ172+ζ32ζ17-ζ1715-ζ179-ζ178-ζ172 | complex faithful |
►On 51 points
Generators in S51
(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 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51)
(2 3 5 9 17 33 14 27)(4 7 13 25 49 46 40 28)(6 11 21 41 30 8 15 29)(10 19 37 22 43 34 16 31)(12 23 45 38 24 47 42 32)(18 35)(20 39 26 51 50 48 44 36)
G:=sub<Sym(51)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51), (2,3,5,9,17,33,14,27)(4,7,13,25,49,46,40,28)(6,11,21,41,30,8,15,29)(10,19,37,22,43,34,16,31)(12,23,45,38,24,47,42,32)(18,35)(20,39,26,51,50,48,44,36)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51), (2,3,5,9,17,33,14,27)(4,7,13,25,49,46,40,28)(6,11,21,41,30,8,15,29)(10,19,37,22,43,34,16,31)(12,23,45,38,24,47,42,32)(18,35)(20,39,26,51,50,48,44,36) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51)], [(2,3,5,9,17,33,14,27),(4,7,13,25,49,46,40,28),(6,11,21,41,30,8,15,29),(10,19,37,22,43,34,16,31),(12,23,45,38,24,47,42,32),(18,35),(20,39,26,51,50,48,44,36)]])
Matrix representation of C51⋊C8 ►in GL8(𝔽2)
| 1 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 1 | 0 | 0 | 1 | 1 |
| 1 | 0 | 1 | 1 | 1 | 0 | 1 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 1 | 1 | 0 | 0 | 1 |
| 0 | 0 | 1 | 1 | 0 | 1 | 0 | 1 |
| 1 | 0 | 0 | 0 | 1 | 0 | 0 | 1 |
| 1 | 0 | 0 | 1 | 0 | 0 | 0 | 1 |
| 0 | 1 | 1 | 1 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 1 | 1 | 0 | 1 | 1 |
| 0 | 1 | 1 | 1 | 1 | 1 | 0 | 0 |
| 0 | 1 | 1 | 1 | 1 | 1 | 1 | 0 |
| 0 | 1 | 1 | 1 | 1 | 0 | 1 | 1 |
| 0 | 0 | 1 | 1 | 0 | 1 | 1 | 1 |
| 0 | 0 | 1 | 0 | 1 | 0 | 1 | 0 |
G:=sub<GL(8,GF(2))| [1,0,0,1,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,1,1,0,1,0,0,0,1,1,0,1,1,0,0,0,0,1,0,1,0,1,0,0,0,0,0,0,1,0,1,0,1,1,1,0,0,0,0,1,1,1,1,1,1,1],[1,0,0,0,0,0,0,0,0,1,0,1,1,1,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,1,1,1,1,0,1,0,1,0,1,1,0,1,0,0,0,1,0,1,1,1,1,1,0,1,0,0,1,1,0] >;
C51⋊C8 in GAP, Magma, Sage, TeX
C_{51}\rtimes C_8 % in TeX
G:=Group("C51:C8"); // GroupNames label
G:=SmallGroup(408,34);
// by ID
G=gap.SmallGroup(408,34);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-17,30,26,323,1204,3909,2414]);
// Polycyclic
G:=Group<a,b|a^51=b^8=1,b*a*b^-1=a^26>;
// generators/relations
Export
Subgroup lattice of C51⋊C8 in TeX
Character table of C51⋊C8 in TeX