Subgroup ($H$) information
| Description: | $S_4$ |
| Order: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Index: | \(4802\)\(\medspace = 2 \cdot 7^{4} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$ab^{3}c^{2}d^{10}f^{3}, b^{2}c^{7}d^{7}e^{5}f^{3}, c^{7}d^{4}, d^{7}ef$
|
| Derived length: | $3$ |
The subgroup is nonabelian, monomial (hence solvable), and rational.
Ambient group ($G$) information
| Description: | $D_7\times C_7^3:S_4$ |
| Order: | \(115248\)\(\medspace = 2^{4} \cdot 3 \cdot 7^{4} \) |
| Exponent: | \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
| Derived length: | $4$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_7^3.(C_7\times A_4).C_6^2.C_2$ |
| $\operatorname{Aut}(H)$ | $S_4$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| $W$ | $S_4$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
Related subgroups
| Centralizer: | $D_7$ | ||
| Normalizer: | $D_7\times S_4$ | ||
| Normal closure: | $C_7^3:S_4$ | ||
| Core: | $C_1$ | ||
| Minimal over-subgroups: | $C_7^3:S_4$ | $C_7\times S_4$ | $C_2\times S_4$ |
| Maximal under-subgroups: | $A_4$ | $D_4$ | $S_3$ |
Other information
| Number of subgroups in this conjugacy class | $343$ |
| Möbius function | $-7$ |
| Projective image | $D_7\times C_7^3:S_4$ |