Subgroup ($H$) information
| Description: | $C_{124}.C_4$ |
| Order: | \(496\)\(\medspace = 2^{4} \cdot 31 \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(248\)\(\medspace = 2^{3} \cdot 31 \) |
| Generators: |
$b^{4}, c^{2}, a^{2}b^{6}c^{31}, c^{31}, ab$
|
| Derived length: | $2$ |
The subgroup is normal, a semidirect factor, nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and hyperelementary for $p = 2$.
Ambient group ($G$) information
| Description: | $(C_2\times C_{248}):C_4$ |
| Order: | \(1984\)\(\medspace = 2^{6} \cdot 31 \) |
| 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: | $C_4$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Automorphism Group: | $C_2$, of order \(2\) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group) and a $p$-group.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{62}.C_{30}.C_2^6.C_2$ |
| $\operatorname{Aut}(H)$ | $C_{62}.C_{30}.C_2^3$ |
| $\card{\operatorname{res}(S)}$ | \(14880\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \cdot 31 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(8\)\(\medspace = 2^{3} \) |
| $W$ | $C_{31}:D_4$, of order \(248\)\(\medspace = 2^{3} \cdot 31 \) |
Related subgroups
Other information
| Möbius function | $0$ |
| Projective image | $C_{62}.C_4^2$ |