Subgroup ($H$) information
| Description: | $C_9$ |
| Order: | \(9\)\(\medspace = 3^{2} \) |
| Index: | \(88\)\(\medspace = 2^{3} \cdot 11 \) |
| Exponent: | \(9\)\(\medspace = 3^{2} \) |
| Generators: |
$\left(\begin{array}{rr}
162 & 0 \\
0 & 162
\end{array}\right), \left(\begin{array}{rr}
92 & 0 \\
0 & 92
\end{array}\right)$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), a direct factor, cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), central, a $3$-Sylow subgroup (hence a Hall subgroup), and a $p$-group.
Ambient group ($G$) information
| Description: | $C_{99}:D_4$ |
| Order: | \(792\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 11 \) |
| Exponent: | \(396\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 11 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
Quotient group ($Q$) structure
| Description: | $C_{11}:D_4$ |
| Order: | \(88\)\(\medspace = 2^{3} \cdot 11 \) |
| Exponent: | \(44\)\(\medspace = 2^{2} \cdot 11 \) |
| Automorphism Group: | $C_2^2\times F_{11}$, of order \(440\)\(\medspace = 2^{3} \cdot 5 \cdot 11 \) |
| Outer Automorphisms: | $C_{10}$, of order \(10\)\(\medspace = 2 \cdot 5 \) |
| 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2\times C_{11}:(C_2^2\times C_{30})$ |
| $\operatorname{Aut}(H)$ | $C_6$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_6$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(440\)\(\medspace = 2^{3} \cdot 5 \cdot 11 \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_{99}:D_4$ | |||
| Normalizer: | $C_{99}:D_4$ | |||
| Complements: | $C_{11}:D_4$ | |||
| Minimal over-subgroups: | $C_{99}$ | $C_{18}$ | $C_{18}$ | $C_{18}$ |
| Maximal under-subgroups: | $C_3$ |
Other information
| Möbius function | $0$ |
| Projective image | $C_{11}:D_4$ |