Subgroup ($H$) information
| Description: | $C_7:C_3$ |
| Order: | \(21\)\(\medspace = 3 \cdot 7 \) |
| Index: | \(32\)\(\medspace = 2^{5} \) |
| Exponent: | \(21\)\(\medspace = 3 \cdot 7 \) |
| Generators: |
$\langle(2,4,3,6,8,7,5), (2,4,6)(3,7,8)\rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian, a Hall subgroup, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 3$.
Ambient group ($G$) information
| Description: | $F_8:C_{12}$ |
| Order: | \(672\)\(\medspace = 2^{5} \cdot 3 \cdot 7 \) |
| Exponent: | \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
| Derived length: | $3$ |
The ambient group is nonabelian, monomial (hence solvable), and an A-group.
Quotient set structure
Since this subgroup has trivial core, the ambient group $G$ acts faithfully and transitively on the set of cosets of $H$. The resulting permutation representation is isomorphic to 32T34615.
Automorphism information
While the subgroup $H$ is not characteristic, the stabilizer $S$ of $H$ in the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : S \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphisms $\operatorname{Inn}(G) \cap S$ is the Weyl group $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $F_8:C_6$, of order \(336\)\(\medspace = 2^{4} \cdot 3 \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $F_7$, of order \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
| $\operatorname{res}(S)$ | $C_7:C_3$, of order \(21\)\(\medspace = 3 \cdot 7 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(2\) |
| $W$ | $C_7:C_3$, of order \(21\)\(\medspace = 3 \cdot 7 \) |
Related subgroups
| Centralizer: | $C_4$ | |
| Normalizer: | $C_7:C_{12}$ | |
| Normal closure: | $F_8:C_3$ | |
| Core: | $C_1$ | |
| Minimal over-subgroups: | $F_8:C_3$ | $C_7:C_6$ |
| Maximal under-subgroups: | $C_7$ | $C_3$ |
Other information
| Number of subgroups in this conjugacy class | $8$ |
| Möbius function | $0$ |
| Projective image | $F_8:C_{12}$ |