p-group, cyclic, elementary abelian, simple, monomial
Aliases: C7, also denoted Z7, SmallGroup(7,1)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
| C1 — C7 |
| C1 — C7 |
| C1 — C7 |
| C1 — C7 |
Generators and relations for C7
G = < a | a7=1 >
Character table of C7
| class | 1 | 7A | 7B | 7C | 7D | 7E | 7F | |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | ζ76 | ζ72 | ζ73 | ζ74 | ζ75 | ζ7 | linear of order 7 faithful |
| ρ3 | 1 | ζ75 | ζ74 | ζ76 | ζ7 | ζ73 | ζ72 | linear of order 7 faithful |
| ρ4 | 1 | ζ74 | ζ76 | ζ72 | ζ75 | ζ7 | ζ73 | linear of order 7 faithful |
| ρ5 | 1 | ζ73 | ζ7 | ζ75 | ζ72 | ζ76 | ζ74 | linear of order 7 faithful |
| ρ6 | 1 | ζ72 | ζ73 | ζ7 | ζ76 | ζ74 | ζ75 | linear of order 7 faithful |
| ρ7 | 1 | ζ7 | ζ75 | ζ74 | ζ73 | ζ72 | ζ76 | linear of order 7 faithful |
►Regular action on 7 points - transitive group 7T1
Generators in S7
(1 2 3 4 5 6 7)
G:=sub<Sym(7)| (1,2,3,4,5,6,7)>;
G:=Group( (1,2,3,4,5,6,7) );
G=PermutationGroup([[(1,2,3,4,5,6,7)]])
G:=TransitiveGroup(7,1);
C7 is a maximal subgroup of
D7 C7⋊C3 C49 F8 C29⋊C7 C43⋊C7 C71⋊C7
C7 is a maximal quotient of C49 F8 C29⋊C7 C43⋊C7 C71⋊C7
| action | f(x) | Disc(f) |
|---|---|---|
| 7T1 | x7-x6-12x5+7x4+28x3-14x2-9x-1 | 172·296 |
Matrix representation of C7 ►in GL1(𝔽29) generated by
| 7 |
G:=sub<GL(1,GF(29))| [7] >;
C7 in GAP, Magma, Sage, TeX
C_7
% in TeX
G:=Group("C7"); // GroupNames label
G:=SmallGroup(7,1);
// by ID
G=gap.SmallGroup(7,1);
# by ID
G:=PCGroup([1,-7]:ExponentLimit:=1);
// Polycyclic
G:=Group<a|a^7=1>;
// generators/relations
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