Subgroup ($H$) information
| Description: | $C_2^2\times A_4$ |
| Order: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Index: | \(2\) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$\left(\begin{array}{rr}
7 & 0 \\
0 & 7
\end{array}\right), \left(\begin{array}{rr}
5 & 4 \\
0 & 5
\end{array}\right), \left(\begin{array}{rr}
3 & 0 \\
0 & 3
\end{array}\right), \left(\begin{array}{rr}
0 & 3 \\
5 & 7
\end{array}\right), \left(\begin{array}{rr}
5 & 0 \\
4 & 5
\end{array}\right)$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), maximal, a semidirect factor, nonabelian, monomial (hence solvable), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $C_2^2\times S_4$ |
| Order: | \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Derived length: | $3$ |
The ambient group is nonabelian, monomial (hence solvable), and rational.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $S_4^2$, of order \(576\)\(\medspace = 2^{6} \cdot 3^{2} \) |
| $\operatorname{Aut}(H)$ | $S_3\times S_4$, of order \(144\)\(\medspace = 2^{4} \cdot 3^{2} \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $S_3\times S_4$, of order \(144\)\(\medspace = 2^{4} \cdot 3^{2} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(4\)\(\medspace = 2^{2} \) |
| $W$ | $S_4$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
Related subgroups
| Centralizer: | $C_2^2$ | ||
| Normalizer: | $C_2^2\times S_4$ | ||
| Complements: | $C_2$ | ||
| Minimal over-subgroups: | $C_2^2\times S_4$ | ||
| Maximal under-subgroups: | $C_2\times A_4$ | $C_2^4$ | $C_2\times C_6$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $-1$ |
| Projective image | $S_4$ |