Subgroup ($H$) information
| Description: | $(C_4\times C_8).\GL(2,\mathbb{Z}/4)$ |
| Order: | \(3072\)\(\medspace = 2^{10} \cdot 3 \) |
| Index: | \(2\) |
| Exponent: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Generators: |
$\left(\begin{array}{rr}
7 & 15 \\
30 & 9
\end{array}\right), \left(\begin{array}{rr}
12 & 27 \\
25 & 19
\end{array}\right), \left(\begin{array}{rr}
11 & 8 \\
8 & 3
\end{array}\right), \left(\begin{array}{rr}
9 & 0 \\
0 & 9
\end{array}\right), \left(\begin{array}{rr}
17 & 16 \\
0 & 17
\end{array}\right), \left(\begin{array}{rr}
31 & 14 \\
26 & 21
\end{array}\right), \left(\begin{array}{rr}
13 & 24 \\
8 & 5
\end{array}\right), \left(\begin{array}{rr}
17 & 0 \\
0 & 17
\end{array}\right), \left(\begin{array}{rr}
25 & 16 \\
16 & 9
\end{array}\right), \left(\begin{array}{rr}
1 & 16 \\
16 & 1
\end{array}\right), \left(\begin{array}{rr}
19 & 16 \\
0 & 19
\end{array}\right)$
|
| Derived length: | $3$ |
The subgroup is characteristic (hence normal), maximal, a semidirect factor, nonabelian, and solvable. Whether it is monomial has not been computed.
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.
Quotient group ($Q$) structure
| Description: | $C_2$ |
| Order: | \(2\) |
| Exponent: | \(2\) |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group, simple, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_2^3\times A_4).C_2^6.C_2^5$ |
| $\operatorname{Aut}(H)$ | $(C_4\times A_4).C_2^5.C_2^6$ |
| $W$ | $C_2^3:D_{24}$, of order \(384\)\(\medspace = 2^{7} \cdot 3 \) |
Related subgroups
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_2^3:D_{24}$ |