Subgroup ($H$) information
| Description: | $C_1$ |
| Order: | $1$ |
| Index: | \(5040\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \) |
| Exponent: | $1$ |
| Generators: | |
| Nilpotency class: | $0$ |
| Derived length: | $0$ |
The subgroup is the center (hence characteristic, normal, abelian, central, nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), the Frattini subgroup, the Fitting subgroup, the radical, a direct factor, cyclic (hence elementary (for every $p$), hyperelementary, metacyclic, and a Z-group), stem, a $p$-group (for every $p$), perfect, and rational.
Ambient group ($G$) information
| Description: | $S_7$ |
| Order: | \(5040\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \) |
| Exponent: | \(420\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 7 \) |
| Derived length: | $1$ |
The ambient group is nonabelian, almost simple, nonsolvable, and rational.
Quotient group ($Q$) structure
| Description: | $S_7$ |
| Order: | \(5040\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \) |
| Exponent: | \(420\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 7 \) |
| Automorphism Group: | $S_7$, of order \(5040\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \) |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Nilpotency class: | $-1$ |
| Derived length: | $1$ |
The quotient is nonabelian, almost simple, nonsolvable, and rational.
Automorphism information
Since the subgroup $H$ is characteristic, the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : \operatorname{Aut}(G) \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphism group $\operatorname{Inn}(G)$ is the Weyl group $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $S_7$, of order \(5040\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $C_1$, of order $1$ |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $S_7$ | ||||||
| Normalizer: | $S_7$ | ||||||
| Complements: | $S_7$ | ||||||
| Minimal over-subgroups: | $C_7$ | $C_5$ | $C_3$ | $C_3$ | $C_2$ | $C_2$ | $C_2$ |
Other information
| Möbius function | $2520$ |
| Projective image | $S_7$ |