direct product, metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary
Aliases: C2×C7⋊C3, C14⋊C3, C7⋊2C6, SmallGroup(42,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — C2×C7⋊C3 |
Generators and relations for C2×C7⋊C3
G = < a,b,c | a2=b7=c3=1, ab=ba, ac=ca, cbc-1=b4 >
Character table of C2×C7⋊C3
| class | 1 | 2 | 3A | 3B | 6A | 6B | 7A | 7B | 14A | 14B | |
| size | 1 | 1 | 7 | 7 | 7 | 7 | 3 | 3 | 3 | 3 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 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 | linear of order 3 |
| ρ4 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ5 | 1 | -1 | ζ3 | ζ32 | ζ6 | ζ65 | 1 | 1 | -1 | -1 | linear of order 6 |
| ρ6 | 1 | -1 | ζ32 | ζ3 | ζ65 | ζ6 | 1 | 1 | -1 | -1 | linear of order 6 |
| ρ7 | 3 | 3 | 0 | 0 | 0 | 0 | -1+√-7/2 | -1-√-7/2 | -1-√-7/2 | -1+√-7/2 | complex lifted from C7⋊C3 |
| ρ8 | 3 | -3 | 0 | 0 | 0 | 0 | -1+√-7/2 | -1-√-7/2 | 1+√-7/2 | 1-√-7/2 | complex faithful |
| ρ9 | 3 | -3 | 0 | 0 | 0 | 0 | -1-√-7/2 | -1+√-7/2 | 1-√-7/2 | 1+√-7/2 | complex faithful |
| ρ10 | 3 | 3 | 0 | 0 | 0 | 0 | -1-√-7/2 | -1+√-7/2 | -1+√-7/2 | -1-√-7/2 | complex lifted from C7⋊C3 |
►On 14 points - transitive group 14T5
Generators in S14
(1 8)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)
(1 2 3 4 5 6 7)(8 9 10 11 12 13 14)
(2 3 5)(4 7 6)(9 10 12)(11 14 13)
G:=sub<Sym(14)| (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14), (1,2,3,4,5,6,7)(8,9,10,11,12,13,14), (2,3,5)(4,7,6)(9,10,12)(11,14,13)>;
G:=Group( (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14), (1,2,3,4,5,6,7)(8,9,10,11,12,13,14), (2,3,5)(4,7,6)(9,10,12)(11,14,13) );
G=PermutationGroup([[(1,8),(2,9),(3,10),(4,11),(5,12),(6,13),(7,14)], [(1,2,3,4,5,6,7),(8,9,10,11,12,13,14)], [(2,3,5),(4,7,6),(9,10,12),(11,14,13)]])
G:=TransitiveGroup(14,5);
C2×C7⋊C3 is a maximal subgroup of
C7⋊C12 C14.A4 C7⋊3F7 C7⋊4F7 SL2(𝔽7)
C2×C7⋊C3 is a maximal quotient of C7⋊3F7 C7⋊4F7
| action | f(x) | Disc(f) |
|---|---|---|
| 14T5 | x14+28x12+308x10+1680x8+4704x6+6272x4+3136x2+484 | -240·720·1114 |
Matrix representation of C2×C7⋊C3 ►in GL3(𝔽11) generated by
| 10 | 0 | 0 |
| 0 | 10 | 0 |
| 0 | 0 | 10 |
| 0 | 8 | 7 |
| 10 | 3 | 3 |
| 0 | 8 | 3 |
| 1 | 8 | 8 |
| 0 | 3 | 8 |
| 0 | 8 | 7 |
G:=sub<GL(3,GF(11))| [10,0,0,0,10,0,0,0,10],[0,10,0,8,3,8,7,3,3],[1,0,0,8,3,8,8,8,7] >;
C2×C7⋊C3 in GAP, Magma, Sage, TeX
C_2\times C_7\rtimes C_3
% in TeX
G:=Group("C2xC7:C3"); // GroupNames label
G:=SmallGroup(42,2);
// by ID
G=gap.SmallGroup(42,2);
# by ID
G:=PCGroup([3,-2,-3,-7,59]);
// Polycyclic
G:=Group<a,b,c|a^2=b^7=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^4>;
// generators/relations
Export
Subgroup lattice of C2×C7⋊C3 in TeX
Character table of C2×C7⋊C3 in TeX