Subgroup ($H$) information
| Description: | $C_{14}$ |
| Order: | \(14\)\(\medspace = 2 \cdot 7 \) |
| Index: | \(81\)\(\medspace = 3^{4} \) |
| Exponent: | \(14\)\(\medspace = 2 \cdot 7 \) |
| Generators: |
$d^{21}, d^{6}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), a direct factor, cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,7$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), central, and a Hall subgroup.
Ambient group ($G$) information
| Description: | $\He_3\times C_{42}$ |
| Order: | \(1134\)\(\medspace = 2 \cdot 3^{4} \cdot 7 \) |
| Exponent: | \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The ambient group is nonabelian, elementary for $p = 3$ (hence nilpotent, solvable, supersolvable, monomial, and hyperelementary), and metabelian.
Quotient group ($Q$) structure
| Description: | $C_3\times \He_3$ |
| Order: | \(81\)\(\medspace = 3^{4} \) |
| Exponent: | \(3\) |
| Automorphism Group: | $C_3^4:(S_3\times \GL(2,3))$, of order \(23328\)\(\medspace = 2^{5} \cdot 3^{6} \) |
| Outer Automorphisms: | $S_3\times C_3^2:\GL(2,3)$, of order \(2592\)\(\medspace = 2^{5} \cdot 3^{4} \) |
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The quotient is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metabelian.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_6\times C_3^4.Q_8.S_3^2$ |
| $\operatorname{Aut}(H)$ | $C_6$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_6$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(23328\)\(\medspace = 2^{5} \cdot 3^{6} \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $\He_3\times C_{42}$ | ||
| Normalizer: | $\He_3\times C_{42}$ | ||
| Complements: | $C_3\times \He_3$ | ||
| Minimal over-subgroups: | $C_{42}$ | $C_{42}$ | $C_{42}$ |
| Maximal under-subgroups: | $C_7$ | $C_2$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $0$ |
| Projective image | $C_3\times \He_3$ |