Subgroup ($H$) information
| Description: | $C_2^3$ |
| Order: | \(8\)\(\medspace = 2^{3} \) |
| Index: | \(248\)\(\medspace = 2^{3} \cdot 31 \) |
| Exponent: | \(2\) |
| Generators: |
$a^{2}, c^{62}, b^{2}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), and rational.
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: | $C_{31}:D_4$ |
| Order: | \(248\)\(\medspace = 2^{3} \cdot 31 \) |
| Exponent: | \(124\)\(\medspace = 2^{2} \cdot 31 \) |
| Automorphism Group: | $C_2^2\times F_{31}$, of order \(3720\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 31 \) |
| Outer Automorphisms: | $C_{30}$, of order \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{62}.C_{30}.C_2^6.C_2$ |
| $\operatorname{Aut}(H)$ | $\PSL(2,7)$, of order \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(59520\)\(\medspace = 2^{7} \cdot 3 \cdot 5 \cdot 31 \) |
| $W$ | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
Related subgroups
| Centralizer: | $C_2^2\times C_{124}$ | |||
| Normalizer: | $C_{124}.C_4^2$ | |||
| Minimal over-subgroups: | $C_2^2\times C_{62}$ | $C_2^2\times C_4$ | $C_2^2:C_4$ | $C_2^2:C_4$ |
| Maximal under-subgroups: | $C_2^2$ | $C_2^2$ | $C_2^2$ | $C_2^2$ |
Other information
| Möbius function | $0$ |
| Projective image | $C_{62}.C_4^2$ |