Subgroup ($H$) information
| Description: | $C_2\times C_{62}$ |
| Order: | \(124\)\(\medspace = 2^{2} \cdot 31 \) |
| Index: | \(16\)\(\medspace = 2^{4} \) |
| Exponent: | \(62\)\(\medspace = 2 \cdot 31 \) |
| Generators: |
$b^{4}, c^{31}, c^{2}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), elementary for $p = 2$ (hence hyperelementary), and metacyclic.
Ambient group ($G$) information
| Description: | $C_{124}.C_4^2$ |
| 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: | $C_4:C_4$ |
| Order: | \(16\)\(\medspace = 2^{4} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Automorphism Group: | $C_2^2\wr C_2$, of order \(32\)\(\medspace = 2^{5} \) |
| Outer Automorphisms: | $D_4$, of order \(8\)\(\medspace = 2^{3} \) |
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The quotient is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metacyclic (hence 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{62}.C_{30}.C_2^6.C_2$ |
| $\operatorname{Aut}(H)$ | $S_3\times C_{30}$, of order \(180\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_{30}$, of order \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(7936\)\(\medspace = 2^{8} \cdot 31 \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
Other information
| Möbius function | $0$ |
| Projective image | $C_{62}.D_4$ |