Subgroup ($H$) information
| Description: | $C_7:C_{12}$ |
| Order: | \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
| Index: | \(40\)\(\medspace = 2^{3} \cdot 5 \) |
| Exponent: | \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
| Generators: |
$\langle(9,13,11,12), (9,11)(12,13), (2,8,7,4,5,6,3), (3,6,4)(5,8,7)\rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 3$.
Ambient group ($G$) information
| Description: | $F_5\times F_8:C_3$ |
| Order: | \(3360\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \cdot 7 \) |
| Exponent: | \(420\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 7 \) |
| 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 40T2718.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $F_5\times F_8:C_3$, of order \(3360\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $C_2\times F_7$, of order \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
| $\operatorname{res}(S)$ | $C_7:C_3$, of order \(21\)\(\medspace = 3 \cdot 7 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(4\)\(\medspace = 2^{2} \) |
| $W$ | $C_7:C_3$, of order \(21\)\(\medspace = 3 \cdot 7 \) |
Related subgroups
| Centralizer: | $C_4$ | ||
| Normalizer: | $C_7:C_{12}$ | ||
| Normal closure: | $F_5\times F_8:C_3$ | ||
| Core: | $C_1$ | ||
| Minimal over-subgroups: | $F_8:C_{12}$ | $C_{35}:C_{12}$ | |
| Maximal under-subgroups: | $C_7:C_6$ | $C_{28}$ | $C_{12}$ |
Other information
| Number of subgroups in this conjugacy class | $40$ |
| Möbius function | $1$ |
| Projective image | $F_5\times F_8:C_3$ |