Subgroup ($H$) information
| Description: | $C_1$ |
| Order: | $1$ |
| Index: | \(992\)\(\medspace = 2^{5} \cdot 31 \) |
| Exponent: | $1$ |
| Generators: | |
| Nilpotency class: | $0$ |
| Derived length: | $0$ |
The subgroup is characteristic (hence normal), a direct factor, cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary (for every $p$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), stem (hence central), a $p$-group (for every $p$), perfect, and rational.
Ambient group ($G$) information
| Description: | $C_{124}.D_4$ |
| Order: | \(992\)\(\medspace = 2^{5} \cdot 31 \) |
| Exponent: | \(248\)\(\medspace = 2^{3} \cdot 31 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
Quotient group ($Q$) structure
| Description: | $C_{124}.D_4$ |
| Order: | \(992\)\(\medspace = 2^{5} \cdot 31 \) |
| Exponent: | \(248\)\(\medspace = 2^{3} \cdot 31 \) |
| Automorphism Group: | $(C_{62}\times D_4).C_{30}.C_2^3$ |
| Outer Automorphisms: | $D_4\times C_{30}$, of order \(240\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_{62}\times D_4).C_{30}.C_2^3$ |
| $\operatorname{Aut}(H)$ | $C_1$, of order $1$ |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_{124}.D_4$ | ||
| Normalizer: | $C_{124}.D_4$ | ||
| Complements: | $C_{124}.D_4$ | ||
| Minimal over-subgroups: | $C_{31}$ | $C_2$ | $C_2$ |
Other information
| Möbius function | $0$ |
| Projective image | $C_{124}.D_4$ |