Subgroup ($H$) information
| Description: | $C_7:C_{28}$ |
| Order: | \(196\)\(\medspace = 2^{2} \cdot 7^{2} \) |
| Index: | \(3\) |
| Exponent: | \(28\)\(\medspace = 2^{2} \cdot 7 \) |
| Generators: |
$a^{3}, a^{6}, c, b$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), maximal, a semidirect factor, nonabelian, a Hall subgroup, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and an A-group.
Ambient group ($G$) information
| Description: | $C_7^2:C_{12}$ |
| Order: | \(588\)\(\medspace = 2^{2} \cdot 3 \cdot 7^{2} \) |
| Exponent: | \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
Quotient group ($Q$) structure
| Description: | $C_3$ |
| Order: | \(3\) |
| Exponent: | \(3\) |
| Automorphism Group: | $C_2$, of order \(2\) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group, and simple.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2\times F_7^2$, of order \(3528\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 7^{2} \) |
| $\operatorname{Aut}(H)$ | $C_{14}:C_6^2$, of order \(504\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 7 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_{14}:C_6^2$, of order \(504\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 7 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(7\) |
| $W$ | $F_7$, of order \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
Related subgroups
| Centralizer: | $C_{14}$ | ||
| Normalizer: | $C_7^2:C_{12}$ | ||
| Complements: | $C_3$ | ||
| Minimal over-subgroups: | $C_7^2:C_{12}$ | ||
| Maximal under-subgroups: | $C_7\times C_{14}$ | $C_7:C_4$ | $C_{28}$ |
Other information
| Möbius function | $-1$ |
| Projective image | $C_7:F_7$ |