Subgroup ($H$) information
| Description: | $(C_4\times C_8).\GL(2,\mathbb{Z}/4)$ |
| Order: | \(3072\)\(\medspace = 2^{10} \cdot 3 \) |
| Index: | \(16\)\(\medspace = 2^{4} \) |
| Exponent: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Generators: |
$\left(\begin{array}{rr}
3 & 29 \\
4 & 29
\end{array}\right), \left(\begin{array}{rr}
24 & 11 \\
21 & 7
\end{array}\right), \left(\begin{array}{rr}
7 & 16 \\
16 & 23
\end{array}\right), \left(\begin{array}{rr}
9 & 0 \\
0 & 9
\end{array}\right), \left(\begin{array}{rr}
17 & 0 \\
16 & 17
\end{array}\right), \left(\begin{array}{rr}
27 & 30 \\
2 & 1
\end{array}\right), \left(\begin{array}{rr}
3 & 8 \\
24 & 11
\end{array}\right), \left(\begin{array}{rr}
17 & 0 \\
0 & 17
\end{array}\right), \left(\begin{array}{rr}
25 & 16 \\
16 & 9
\end{array}\right), \left(\begin{array}{rr}
17 & 16 \\
0 & 17
\end{array}\right), \left(\begin{array}{rr}
29 & 0 \\
0 & 29
\end{array}\right)$
|
| Derived length: | $3$ |
The subgroup is nonabelian and solvable. Whether it is monomial has not been computed.
Ambient group ($G$) information
| Description: | $(C_4^3\times C_8).\GL(2,\mathbb{Z}/4)$ |
| Order: | \(49152\)\(\medspace = 2^{14} \cdot 3 \) |
| Exponent: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Derived length: | $4$ |
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_5^4:(C_2\times Q_8).Q_8$, of order \(6291456\)\(\medspace = 2^{21} \cdot 3 \) |
| $\operatorname{Aut}(H)$ | $(C_4\times A_4).C_2^5.C_2^6$ |
| $W$ | $C_8:S_4$, of order \(192\)\(\medspace = 2^{6} \cdot 3 \) |
Related subgroups
| Centralizer: | $C_2\times C_8$ |
| Normalizer: | $(C_4\times C_8).\GL(2,\mathbb{Z}/4)$ |
| Normal closure: | $(C_4^3\times C_8).\GL(2,\mathbb{Z}/4)$ |
| Core: | $C_2^3\times C_4\times C_8$ |
Other information
| Number of subgroups in this autjugacy class | $16$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | not computed |