Subgroup ($H$) information
| Description: | $C_7$ |
| Order: | \(7\) |
| Index: | \(243\)\(\medspace = 3^{5} \) |
| Exponent: | \(7\) |
| Generators: |
$d^{9}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), a direct factor, cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), central, a $7$-Sylow subgroup (hence a Hall subgroup), a $p$-group, and simple.
Ambient group ($G$) information
| Description: | $C_3^3\times C_{63}$ |
| Order: | \(1701\)\(\medspace = 3^{5} \cdot 7 \) |
| Exponent: | \(63\)\(\medspace = 3^{2} \cdot 7 \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The ambient group is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group) and elementary for $p = 3$ (hence hyperelementary).
Quotient group ($Q$) structure
| Description: | $C_3^3\times C_9$ |
| Order: | \(243\)\(\medspace = 3^{5} \) |
| Exponent: | \(9\)\(\medspace = 3^{2} \) |
| Automorphism Group: | $C_3^4.(C_2\times C_3^3:\GL(3,3))$, of order \(49128768\)\(\medspace = 2^{6} \cdot 3^{10} \cdot 13 \) |
| Outer Automorphisms: | $C_3^4.(C_2\times C_3^3:\GL(3,3))$, of order \(49128768\)\(\medspace = 2^{6} \cdot 3^{10} \cdot 13 \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group) and a $p$-group (hence elementary and hyperelementary).
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^4.C_3^4.C_2^3.\SL(3,3)$, of order \(294772608\)\(\medspace = 2^{7} \cdot 3^{11} \cdot 13 \) |
| $\operatorname{Aut}(H)$ | $C_6$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_6$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(49128768\)\(\medspace = 2^{6} \cdot 3^{10} \cdot 13 \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_3^3\times C_{63}$ | |
| Normalizer: | $C_3^3\times C_{63}$ | |
| Complements: | $C_3^3\times C_9$ | |
| Minimal over-subgroups: | $C_{21}$ | $C_{21}$ |
| Maximal under-subgroups: | $C_1$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $0$ |
| Projective image | $C_3^3\times C_9$ |