Subgroup ($H$) information
| Description: | $C_4$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Index: | \(28812\)\(\medspace = 2^{2} \cdot 3 \cdot 7^{4} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Generators: |
$ab^{3}c^{2}d^{3}ef^{5}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group) and a $p$-group.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_7^3.(C_7\times A_4).C_6^2.C_2$ |
| $\operatorname{Aut}(H)$ | $C_2$, of order \(2\) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
| Centralizer: | $C_4\times D_7$ | |||||
| Normalizer: | $D_4\times D_7$ | |||||
| Normal closure: | $C_7^3:S_4$ | |||||
| Core: | $C_1$ | |||||
| Minimal over-subgroups: | $C_7^2:C_4$ | $C_{28}$ | $C_7:C_4$ | $C_2\times C_4$ | $D_4$ | $D_4$ |
| Maximal under-subgroups: | $C_2$ |
Other information
| Number of subgroups in this conjugacy class | $1029$ |
| Möbius function | $0$ |
| Projective image | $D_7\times C_7^3:S_4$ |