Subgroup ($H$) information
| Description: | $Q_8:C_{99}$ |
| Order: | \(792\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 11 \) |
| Index: | $1$ |
| Exponent: | \(396\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 11 \) |
| Generators: |
$\left(\begin{array}{rr}
181 & 0 \\
0 & 181
\end{array}\right), \left(\begin{array}{rr}
164 & 101 \\
45 & 35
\end{array}\right), \left(\begin{array}{rr}
198 & 0 \\
0 & 198
\end{array}\right), \left(\begin{array}{rr}
131 & 40 \\
143 & 65
\end{array}\right), \left(\begin{array}{rr}
190 & 50 \\
70 & 9
\end{array}\right), \left(\begin{array}{rr}
27 & 0 \\
0 & 27
\end{array}\right)$
|
| Derived length: | $3$ |
The subgroup is the radical (hence characteristic, normal, and solvable), a direct factor, nonabelian, and a Hall subgroup.
Ambient group ($G$) information
| Description: | $Q_8:C_{99}$ |
| Order: | \(792\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 11 \) |
| Exponent: | \(396\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 11 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and solvable.
Quotient group ($Q$) structure
| Description: | $C_1$ |
| Order: | $1$ |
| Exponent: | $1$ |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $0$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary (for every $p$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group (for every $p$), perfect, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $S_4\times C_{30}$, of order \(720\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \) |
| $\operatorname{Aut}(H)$ | $S_4\times C_{30}$, of order \(720\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \) |
| $W$ | $A_4$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
Related subgroups
| Centralizer: | $C_{66}$ | ||
| Normalizer: | $Q_8:C_{99}$ | ||
| Complements: | $C_1$ | ||
| Maximal under-subgroups: | $Q_8\times C_{33}$ | $C_{198}$ | $Q_8:C_9$ |
Other information
| Möbius function | $1$ |
| Projective image | $A_4$ |