Subgroup ($H$) information
| Description: | $C_2\times C_{48}$ |
| Order: | \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| Index: | \(2\) |
| Exponent: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Generators: |
$\left(\begin{array}{rr}
1 & 0 \\
0 & 96
\end{array}\right), \left(\begin{array}{rr}
48 & 0 \\
0 & 11
\end{array}\right), \left(\begin{array}{rr}
96 & 0 \\
0 & 96
\end{array}\right), \left(\begin{array}{rr}
36 & 0 \\
0 & 36
\end{array}\right), \left(\begin{array}{rr}
22 & 0 \\
0 & 75
\end{array}\right), \left(\begin{array}{rr}
73 & 0 \\
0 & 24
\end{array}\right)$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), maximal, a semidirect factor, abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), elementary for $p = 2$ (hence hyperelementary), and metacyclic.
Ambient group ($G$) information
| Description: | $\OD_{32}:C_6$ |
| Order: | \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| Exponent: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Nilpotency class: | $3$ |
| Derived length: | $2$ |
The ambient group is nonabelian, elementary for $p = 2$ (hence nilpotent, solvable, supersolvable, monomial, and hyperelementary), and metabelian.
Quotient group ($Q$) structure
| Description: | $C_2$ |
| Order: | \(2\) |
| Exponent: | \(2\) |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group, simple, and rational.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_4^2:C_2^3$, of order \(128\)\(\medspace = 2^{7} \) |
| $\operatorname{Aut}(H)$ | $D_4:C_2^3$, of order \(64\)\(\medspace = 2^{6} \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_2^3\times C_4$, of order \(32\)\(\medspace = 2^{5} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(4\)\(\medspace = 2^{2} \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
| Centralizer: | $C_2\times C_{48}$ | ||
| Normalizer: | $\OD_{32}:C_6$ | ||
| Complements: | $C_2$ | ||
| Minimal over-subgroups: | $\OD_{32}:C_6$ | ||
| Maximal under-subgroups: | $C_2\times C_{24}$ | $C_{48}$ | $C_2\times C_{16}$ |
Other information
| Möbius function | $-1$ |
| Projective image | $D_4$ |