Subgroup ($H$) information
| Description: | $C_2^5.A_4$ |
| Order: | \(384\)\(\medspace = 2^{7} \cdot 3 \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\left(\begin{array}{rr}
7 & 8 \\
0 & 15
\end{array}\right), \left(\begin{array}{rr}
13 & 12 \\
8 & 5
\end{array}\right), \left(\begin{array}{rr}
13 & 0 \\
12 & 5
\end{array}\right), \left(\begin{array}{rr}
9 & 0 \\
0 & 9
\end{array}\right), \left(\begin{array}{rr}
12 & 11 \\
9 & 3
\end{array}\right), \left(\begin{array}{rr}
9 & 0 \\
8 & 9
\end{array}\right), \left(\begin{array}{rr}
9 & 8 \\
0 & 9
\end{array}\right), \left(\begin{array}{rr}
9 & 8 \\
8 & 1
\end{array}\right)$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), nonabelian, monomial (hence solvable), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $C_4^4:C_6$ |
| Order: | \(1536\)\(\medspace = 2^{9} \cdot 3 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, monomial (hence solvable), metabelian, and an A-group.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_4^2:C_3.D_4\times C_2^6.S_4$ |
| $\operatorname{Aut}(H)$ | $C_4^2:C_3.D_4\times \PSL(2,7)$ |
| $\card{\operatorname{res}(\operatorname{Aut}(G))}$ | \(9216\)\(\medspace = 2^{10} \cdot 3^{2} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(64\)\(\medspace = 2^{6} \) |
| $W$ | $C_4^2:C_3$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \) |
Related subgroups
| Centralizer: | $C_2\times C_4^2$ | |||
| Normalizer: | $C_4^4:C_6$ | |||
| Minimal over-subgroups: | $C_4\wr C_3\times C_2^2$ | |||
| Maximal under-subgroups: | $C_2^4.A_4$ | $C_2^4.A_4$ | $C_2^3\times C_4^2$ | $C_2^3\times A_4$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $2$ |
| Projective image | $C_2^4.A_4$ |