Subgroup ($H$) information
| Description: | $C_4^4.(C_8\times A_4)$ |
| Order: | \(24576\)\(\medspace = 2^{13} \cdot 3 \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Generators: |
$\left(\begin{array}{rr}
27 & 26 \\
18 & 9
\end{array}\right), \left(\begin{array}{rr}
17 & 8 \\
24 & 25
\end{array}\right), \left(\begin{array}{rr}
17 & 0 \\
0 & 17
\end{array}\right), \left(\begin{array}{rr}
24 & 31 \\
25 & 7
\end{array}\right), \left(\begin{array}{rr}
9 & 16 \\
16 & 25
\end{array}\right), \left(\begin{array}{rr}
1 & 16 \\
16 & 17
\end{array}\right), \left(\begin{array}{rr}
9 & 24 \\
16 & 25
\end{array}\right), \left(\begin{array}{rr}
19 & 16 \\
0 & 27
\end{array}\right), \left(\begin{array}{rr}
17 & 16 \\
0 & 17
\end{array}\right), \left(\begin{array}{rr}
17 & 0 \\
16 & 17
\end{array}\right), \left(\begin{array}{rr}
9 & 16 \\
8 & 25
\end{array}\right), \left(\begin{array}{rr}
13 & 8 \\
8 & 5
\end{array}\right), \left(\begin{array}{rr}
29 & 24 \\
4 & 21
\end{array}\right), \left(\begin{array}{rr}
29 & 12 \\
8 & 21
\end{array}\right)$
|
| Derived length: | $3$ |
The subgroup is characteristic (hence normal), nonabelian, and solvable. Whether it is a direct factor, a semidirect factor, or 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.
Quotient group ($Q$) structure
| Description: | $C_2^2$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(2\) |
| Automorphism Group: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Outer Automorphisms: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), metacyclic, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | Group of order \(12582912\)\(\medspace = 2^{22} \cdot 3 \) |
| $\operatorname{Aut}(H)$ | $C_4^2.(C_2^4\times A_4).C_2^6.C_2^6.C_2$ |
| $W$ | $C_8^2:(C_2\times D_6)$, of order \(1536\)\(\medspace = 2^{9} \cdot 3 \) |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $C_4^4.C_4^2:S_4$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | not computed |