metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary
Aliases: C7⋊C3, SmallGroup(21,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — C7⋊C3 |
Generators and relations for C7⋊C3
G = < a,b | a7=b3=1, bab-1=a4 >
Character table of C7⋊C3
| class | 1 | 3A | 3B | 7A | 7B | |
| size | 1 | 7 | 7 | 3 | 3 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | ζ32 | ζ3 | 1 | 1 | linear of order 3 |
| ρ3 | 1 | ζ3 | ζ32 | 1 | 1 | linear of order 3 |
| ρ4 | 3 | 0 | 0 | -1-√-7/2 | -1+√-7/2 | complex faithful |
| ρ5 | 3 | 0 | 0 | -1+√-7/2 | -1-√-7/2 | complex faithful |
►On 7 points: primitive - transitive group 7T3
Generators in S7
(1 2 3 4 5 6 7)
(2 3 5)(4 7 6)
G:=sub<Sym(7)| (1,2,3,4,5,6,7), (2,3,5)(4,7,6)>;
G:=Group( (1,2,3,4,5,6,7), (2,3,5)(4,7,6) );
G=PermutationGroup([[(1,2,3,4,5,6,7)], [(2,3,5),(4,7,6)]])
G:=TransitiveGroup(7,3);
►Regular action on 21 points - transitive group 21T2
Generators in S21
(1 2 3 4 5 6 7)(8 9 10 11 12 13 14)(15 16 17 18 19 20 21)
(1 19 10)(2 21 14)(3 16 11)(4 18 8)(5 20 12)(6 15 9)(7 17 13)
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), (1,19,10)(2,21,14)(3,16,11)(4,18,8)(5,20,12)(6,15,9)(7,17,13)>;
G:=Group( (1,2,3,4,5,6,7)(8,9,10,11,12,13,14)(15,16,17,18,19,20,21), (1,19,10)(2,21,14)(3,16,11)(4,18,8)(5,20,12)(6,15,9)(7,17,13) );
G=PermutationGroup([[(1,2,3,4,5,6,7),(8,9,10,11,12,13,14),(15,16,17,18,19,20,21)], [(1,19,10),(2,21,14),(3,16,11),(4,18,8),(5,20,12),(6,15,9),(7,17,13)]])
G:=TransitiveGroup(21,2);
C7⋊C3 is a maximal subgroup of
F7 C7⋊A4 C49⋊C3 C72⋊C3 C72⋊3C3 GL3(𝔽2) AΓL1(𝔽8) C91⋊C3 C91⋊4C3 C133⋊C3 C133⋊4C3
C7⋊C3 is a maximal quotient of
C7⋊C9 C7⋊A4 C49⋊C3 C72⋊C3 C72⋊3C3 AΓL1(𝔽8) C91⋊C3 C91⋊4C3 C133⋊C3 C133⋊4C3
| action | f(x) | Disc(f) |
|---|---|---|
| 7T3 | x7-8x5-2x4+16x3+6x2-6x-2 | 26·734 |
Matrix representation of C7⋊C3 ►in GL3(𝔽2) generated by
| 1 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 1 | 1 |
G:=sub<GL(3,GF(2))| [1,0,1,1,0,0,0,1,0],[1,0,0,0,0,1,0,1,1] >;
C7⋊C3 in GAP, Magma, Sage, TeX
C_7\rtimes C_3
% in TeX
G:=Group("C7:C3"); // GroupNames label
G:=SmallGroup(21,1);
// by ID
G=gap.SmallGroup(21,1);
# by ID
G:=PCGroup([2,-3,-7,25]);
// Polycyclic
G:=Group<a,b|a^7=b^3=1,b*a*b^-1=a^4>;
// generators/relations
Export
Subgroup lattice of C7⋊C3 in TeX
Character table of C7⋊C3 in TeX