Subgroup ($H$) information
| Description: | $D_{12}:S_4$ |
| Order: | \(576\)\(\medspace = 2^{6} \cdot 3^{2} \) |
| Index: | \(3\) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Generators: |
$\left(\begin{array}{rr}
55 & 3 \\
0 & 1
\end{array}\right), \left(\begin{array}{rr}
64 & 21 \\
63 & 43
\end{array}\right), \left(\begin{array}{rr}
1 & 28 \\
0 & 1
\end{array}\right), \left(\begin{array}{rr}
53 & 26 \\
18 & 31
\end{array}\right), \left(\begin{array}{rr}
13 & 0 \\
0 & 13
\end{array}\right), \left(\begin{array}{rr}
43 & 0 \\
42 & 43
\end{array}\right), \left(\begin{array}{rr}
43 & 42 \\
0 & 43
\end{array}\right), \left(\begin{array}{rr}
67 & 12 \\
12 & 31
\end{array}\right)$
|
| Derived length: | $3$ |
The subgroup is characteristic (hence normal), maximal, a direct factor, nonabelian, and monomial (hence solvable).
Ambient group ($G$) information
| Description: | $C_3\times D_{12}:S_4$ |
| Order: | \(1728\)\(\medspace = 2^{6} \cdot 3^{3} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and monomial (hence solvable).
Quotient group ($Q$) structure
| Description: | $C_3$ |
| Order: | \(3\) |
| Exponent: | \(3\) |
| Automorphism Group: | $C_2$, of order \(2\) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| 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, and simple.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^5.D_6^2$, of order \(4608\)\(\medspace = 2^{9} \cdot 3^{2} \) |
| $\operatorname{Aut}(H)$ | $C_2^4:D_6^2$, of order \(2304\)\(\medspace = 2^{8} \cdot 3^{2} \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_2^4:D_6^2$, of order \(2304\)\(\medspace = 2^{8} \cdot 3^{2} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(2\) |
| $W$ | $D_6:S_4$, of order \(288\)\(\medspace = 2^{5} \cdot 3^{2} \) |
Related subgroups
Other information
| Möbius function | $-1$ |
| Projective image | $C_3^2:\GL(2,\mathbb{Z}/4)$ |