Subgroup ($H$) information
| Description: | $C_4^3.(C_4.\GL(2,\mathbb{Z}/4))$ |
| Order: | \(24576\)\(\medspace = 2^{13} \cdot 3 \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Generators: |
$\left(\begin{array}{rr}
3 & 6 \\
6 & 13
\end{array}\right), \left(\begin{array}{rr}
27 & 13 \\
28 & 17
\end{array}\right), \left(\begin{array}{rr}
17 & 0 \\
0 & 17
\end{array}\right), \left(\begin{array}{rr}
13 & 0 \\
0 & 13
\end{array}\right), \left(\begin{array}{rr}
0 & 7 \\
9 & 31
\end{array}\right), \left(\begin{array}{rr}
9 & 0 \\
0 & 9
\end{array}\right), \left(\begin{array}{rr}
9 & 24 \\
16 & 25
\end{array}\right), \left(\begin{array}{rr}
9 & 16 \\
16 & 25
\end{array}\right), \left(\begin{array}{rr}
17 & 16 \\
0 & 17
\end{array}\right), \left(\begin{array}{rr}
17 & 0 \\
16 & 17
\end{array}\right), \left(\begin{array}{rr}
25 & 0 \\
8 & 9
\end{array}\right), \left(\begin{array}{rr}
31 & 24 \\
24 & 31
\end{array}\right), \left(\begin{array}{rr}
21 & 16 \\
4 & 29
\end{array}\right), \left(\begin{array}{rr}
13 & 28 \\
8 & 5
\end{array}\right)$
|
| Derived length: | $4$ |
The subgroup is nonabelian and solvable. Whether it is monomial has not been computed.
Ambient group ($G$) information
| Description: | $C_4^4.C_4^2:S_4$ |
| Order: | \(98304\)\(\medspace = 2^{15} \cdot 3 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | Group of order \(12582912\)\(\medspace = 2^{22} \cdot 3 \) |
| $\operatorname{Aut}(H)$ | $C_5\wr C_2^2:C_4$, of order \(1572864\)\(\medspace = 2^{19} \cdot 3 \) |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $(C_4^3\times C_8).\GL(2,\mathbb{Z}/4)$ |
| Normal closure: | $(C_4^3\times C_8).\GL(2,\mathbb{Z}/4)$ |
| Core: | $C_4^3.C_2^3:C_{24}$ |
Other information
| Number of subgroups in this autjugacy class | $2$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | not computed |