metabelian, supersoluble, monomial, A-group
Aliases: C7⋊5F7, C72⋊3C6, C7⋊C3⋊D7, C7⋊(C3×D7), C7⋊D7⋊1C3, (C7×C7⋊C3)⋊3C2, SmallGroup(294,10)
Series: Derived ►Chief ►Lower central ►Upper central
C1 — C7 — C72 — C7×C7⋊C3 — C7⋊5F7 |
C72 — C7⋊5F7 |
Generators and relations for C7⋊5F7
G = < a,b,c | a7=b7=c6=1, ab=ba, cac-1=a-1, cbc-1=b5 >
Character table of C7⋊5F7
class | 1 | 2 | 3A | 3B | 6A | 6B | 7A | 7B | 7C | 7D | 7E | 7F | 7G | 7H | 7I | 7J | 21A | 21B | 21C | 21D | 21E | 21F | |
size | 1 | 49 | 7 | 7 | 49 | 49 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 14 | 14 | 14 | 14 | 14 | 14 | |
ρ1 | 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 | linear of order 2 |
ρ3 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | linear of order 3 |
ρ4 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | linear of order 3 |
ρ5 | 1 | -1 | ζ3 | ζ32 | ζ6 | ζ65 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | linear of order 6 |
ρ6 | 1 | -1 | ζ32 | ζ3 | ζ65 | ζ6 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | linear of order 6 |
ρ7 | 2 | 0 | 2 | 2 | 0 | 0 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | 2 | ζ76+ζ7 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | orthogonal lifted from D7 |
ρ8 | 2 | 0 | 2 | 2 | 0 | 0 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | 2 | ζ74+ζ73 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | orthogonal lifted from D7 |
ρ9 | 2 | 0 | 2 | 2 | 0 | 0 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | 2 | ζ75+ζ72 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | orthogonal lifted from D7 |
ρ10 | 2 | 0 | -1-√-3 | -1+√-3 | 0 | 0 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | 2 | ζ76+ζ7 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ3ζ74+ζ3ζ73 | ζ3ζ76+ζ3ζ7 | ζ3ζ75+ζ3ζ72 | ζ32ζ76+ζ32ζ7 | ζ32ζ75+ζ32ζ72 | ζ32ζ74+ζ32ζ73 | complex lifted from C3×D7 |
ρ11 | 2 | 0 | -1+√-3 | -1-√-3 | 0 | 0 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | 2 | ζ76+ζ7 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ32ζ74+ζ32ζ73 | ζ32ζ76+ζ32ζ7 | ζ32ζ75+ζ32ζ72 | ζ3ζ76+ζ3ζ7 | ζ3ζ75+ζ3ζ72 | ζ3ζ74+ζ3ζ73 | complex lifted from C3×D7 |
ρ12 | 2 | 0 | -1+√-3 | -1-√-3 | 0 | 0 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | 2 | ζ75+ζ72 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ32ζ76+ζ32ζ7 | ζ32ζ75+ζ32ζ72 | ζ32ζ74+ζ32ζ73 | ζ3ζ75+ζ3ζ72 | ζ3ζ74+ζ3ζ73 | ζ3ζ76+ζ3ζ7 | complex lifted from C3×D7 |
ρ13 | 2 | 0 | -1-√-3 | -1+√-3 | 0 | 0 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | 2 | ζ74+ζ73 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ3ζ75+ζ3ζ72 | ζ3ζ74+ζ3ζ73 | ζ3ζ76+ζ3ζ7 | ζ32ζ74+ζ32ζ73 | ζ32ζ76+ζ32ζ7 | ζ32ζ75+ζ32ζ72 | complex lifted from C3×D7 |
ρ14 | 2 | 0 | -1-√-3 | -1+√-3 | 0 | 0 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | 2 | ζ75+ζ72 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ3ζ76+ζ3ζ7 | ζ3ζ75+ζ3ζ72 | ζ3ζ74+ζ3ζ73 | ζ32ζ75+ζ32ζ72 | ζ32ζ74+ζ32ζ73 | ζ32ζ76+ζ32ζ7 | complex lifted from C3×D7 |
ρ15 | 2 | 0 | -1+√-3 | -1-√-3 | 0 | 0 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | 2 | ζ74+ζ73 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ32ζ75+ζ32ζ72 | ζ32ζ74+ζ32ζ73 | ζ32ζ76+ζ32ζ7 | ζ3ζ74+ζ3ζ73 | ζ3ζ76+ζ3ζ7 | ζ3ζ75+ζ3ζ72 | complex lifted from C3×D7 |
ρ16 | 6 | 0 | 0 | 0 | 0 | 0 | 6 | 6 | 6 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from F7 |
ρ17 | 6 | 0 | 0 | 0 | 0 | 0 | 3ζ75+3ζ72 | 3ζ74+3ζ73 | 3ζ76+3ζ7 | -ζ76-ζ7+1 | ζ76+2ζ74+2ζ73+ζ7 | -1 | -ζ75-ζ72+1 | 2ζ76+ζ75+ζ72+2ζ7 | 2ζ75+ζ74+ζ73+2ζ72 | -ζ74-ζ73+1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
ρ18 | 6 | 0 | 0 | 0 | 0 | 0 | 3ζ75+3ζ72 | 3ζ74+3ζ73 | 3ζ76+3ζ7 | 2ζ76+ζ75+ζ72+2ζ7 | -ζ74-ζ73+1 | -1 | 2ζ75+ζ74+ζ73+2ζ72 | -ζ76-ζ7+1 | -ζ75-ζ72+1 | ζ76+2ζ74+2ζ73+ζ7 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
ρ19 | 6 | 0 | 0 | 0 | 0 | 0 | 3ζ74+3ζ73 | 3ζ76+3ζ7 | 3ζ75+3ζ72 | 2ζ75+ζ74+ζ73+2ζ72 | -ζ76-ζ7+1 | -1 | ζ76+2ζ74+2ζ73+ζ7 | -ζ75-ζ72+1 | -ζ74-ζ73+1 | 2ζ76+ζ75+ζ72+2ζ7 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
ρ20 | 6 | 0 | 0 | 0 | 0 | 0 | 3ζ74+3ζ73 | 3ζ76+3ζ7 | 3ζ75+3ζ72 | -ζ75-ζ72+1 | 2ζ76+ζ75+ζ72+2ζ7 | -1 | -ζ74-ζ73+1 | 2ζ75+ζ74+ζ73+2ζ72 | ζ76+2ζ74+2ζ73+ζ7 | -ζ76-ζ7+1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
ρ21 | 6 | 0 | 0 | 0 | 0 | 0 | 3ζ76+3ζ7 | 3ζ75+3ζ72 | 3ζ74+3ζ73 | -ζ74-ζ73+1 | 2ζ75+ζ74+ζ73+2ζ72 | -1 | -ζ76-ζ7+1 | ζ76+2ζ74+2ζ73+ζ7 | 2ζ76+ζ75+ζ72+2ζ7 | -ζ75-ζ72+1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
ρ22 | 6 | 0 | 0 | 0 | 0 | 0 | 3ζ76+3ζ7 | 3ζ75+3ζ72 | 3ζ74+3ζ73 | ζ76+2ζ74+2ζ73+ζ7 | -ζ75-ζ72+1 | -1 | 2ζ76+ζ75+ζ72+2ζ7 | -ζ74-ζ73+1 | -ζ76-ζ7+1 | 2ζ75+ζ74+ζ73+2ζ72 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
(1 2 3 4 5 6 7)(8 9 10 11 12 13 14)(15 16 17 18 19 20 21)
(1 2 3 4 5 6 7)(8 10 12 14 9 11 13)(15 19 16 20 17 21 18)
(1 8 15)(2 14 16 7 9 21)(3 13 17 6 10 20)(4 12 18 5 11 19)
G:=sub<Sym(21)| (1,2,3,4,5,6,7)(8,9,10,11,12,13,14)(15,16,17,18,19,20,21), (1,2,3,4,5,6,7)(8,10,12,14,9,11,13)(15,19,16,20,17,21,18), (1,8,15)(2,14,16,7,9,21)(3,13,17,6,10,20)(4,12,18,5,11,19)>;
G:=Group( (1,2,3,4,5,6,7)(8,9,10,11,12,13,14)(15,16,17,18,19,20,21), (1,2,3,4,5,6,7)(8,10,12,14,9,11,13)(15,19,16,20,17,21,18), (1,8,15)(2,14,16,7,9,21)(3,13,17,6,10,20)(4,12,18,5,11,19) );
G=PermutationGroup([[(1,2,3,4,5,6,7),(8,9,10,11,12,13,14),(15,16,17,18,19,20,21)], [(1,2,3,4,5,6,7),(8,10,12,14,9,11,13),(15,19,16,20,17,21,18)], [(1,8,15),(2,14,16,7,9,21),(3,13,17,6,10,20),(4,12,18,5,11,19)]])
G:=TransitiveGroup(21,16);
Matrix representation of C7⋊5F7 ►in GL6(𝔽43)
23 | 23 | 0 | 0 | 0 | 0 |
20 | 35 | 0 | 0 | 0 | 0 |
0 | 0 | 23 | 23 | 0 | 0 |
0 | 0 | 20 | 35 | 0 | 0 |
0 | 0 | 0 | 0 | 23 | 23 |
0 | 0 | 0 | 0 | 20 | 35 |
8 | 42 | 0 | 0 | 0 | 0 |
1 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 20 | 35 | 0 | 0 |
0 | 0 | 8 | 42 | 0 | 0 |
0 | 0 | 0 | 0 | 35 | 20 |
0 | 0 | 0 | 0 | 23 | 23 |
0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 0 | 8 | 42 |
1 | 0 | 0 | 0 | 0 | 0 |
8 | 42 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 8 | 42 | 0 | 0 |
G:=sub<GL(6,GF(43))| [23,20,0,0,0,0,23,35,0,0,0,0,0,0,23,20,0,0,0,0,23,35,0,0,0,0,0,0,23,20,0,0,0,0,23,35],[8,1,0,0,0,0,42,0,0,0,0,0,0,0,20,8,0,0,0,0,35,42,0,0,0,0,0,0,35,23,0,0,0,0,20,23],[0,0,1,8,0,0,0,0,0,42,0,0,0,0,0,0,1,8,0,0,0,0,0,42,1,8,0,0,0,0,0,42,0,0,0,0] >;
C7⋊5F7 in GAP, Magma, Sage, TeX
C_7\rtimes_5F_7
% in TeX
G:=Group("C7:5F7");
// GroupNames label
G:=SmallGroup(294,10);
// by ID
G=gap.SmallGroup(294,10);
# by ID
G:=PCGroup([4,-2,-3,-7,-7,434,4035,679]);
// Polycyclic
G:=Group<a,b,c|a^7=b^7=c^6=1,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^5>;
// generators/relations
Export
Subgroup lattice of C7⋊5F7 in TeX
Character table of C7⋊5F7 in TeX