Subgroup ($H$) information
| Description: | $C_2\times C_4.D_{24}$ |
| Order: | \(384\)\(\medspace = 2^{7} \cdot 3 \) |
| Index: | \(16\)\(\medspace = 2^{4} \) |
| Exponent: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Generators: |
$\left(\begin{array}{rr}
29 & 26 \\
30 & 31
\end{array}\right), \left(\begin{array}{rr}
19 & 8 \\
24 & 27
\end{array}\right), \left(\begin{array}{rr}
25 & 16 \\
16 & 9
\end{array}\right), \left(\begin{array}{rr}
15 & 1 \\
0 & 17
\end{array}\right), \left(\begin{array}{rr}
21 & 24 \\
8 & 13
\end{array}\right), \left(\begin{array}{rr}
12 & 11 \\
9 & 19
\end{array}\right), \left(\begin{array}{rr}
17 & 0 \\
0 & 17
\end{array}\right), \left(\begin{array}{rr}
9 & 0 \\
0 & 9
\end{array}\right)$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
Ambient group ($G$) information
| Description: | $C_8^2.\GL(2,\mathbb{Z}/4)$ |
| Order: | \(6144\)\(\medspace = 2^{11} \cdot 3 \) |
| Exponent: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Automorphism information
While the subgroup $H$ is not characteristic, the stabilizer $S$ of $H$ in the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : S \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphisms $\operatorname{Inn}(G) \cap S$ is the Weyl group $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_2^3\times A_4).C_2^6.C_2^5$ |
| $\operatorname{Aut}(H)$ | $C_3:(C_2^2.C_2^6.C_2^3)$ |
| $W$ | $C_2\times D_{24}$, of order \(96\)\(\medspace = 2^{5} \cdot 3 \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $4$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $(C_4\times A_4) . (C_2^2\times C_4)$ |