Subgroup ($H$) information
| Description: | $C_6.\GL(2,\mathbb{Z}/4)$ |
| Order: | \(576\)\(\medspace = 2^{6} \cdot 3^{2} \) |
| Index: | $1$ |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Generators: |
$\left(\begin{array}{rr}
23 & 48 \\
33 & 5
\end{array}\right), \left(\begin{array}{rr}
22 & 21 \\
21 & 1
\end{array}\right), \left(\begin{array}{rr}
1 & 56 \\
0 & 1
\end{array}\right), \left(\begin{array}{rr}
35 & 24 \\
9 & 73
\end{array}\right), \left(\begin{array}{rr}
13 & 0 \\
0 & 13
\end{array}\right), \left(\begin{array}{rr}
1 & 42 \\
42 & 1
\end{array}\right), \left(\begin{array}{rr}
43 & 0 \\
42 & 43
\end{array}\right), \left(\begin{array}{rr}
1 & 12 \\
36 & 13
\end{array}\right)$
|
| Derived length: | $3$ |
The subgroup is the radical (hence characteristic, normal, and solvable), a direct factor, nonabelian, a Hall subgroup, and monomial.
Ambient group ($G$) information
| Description: | $C_6.\GL(2,\mathbb{Z}/4)$ |
| Order: | \(576\)\(\medspace = 2^{6} \cdot 3^{2} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and monomial (hence solvable).
Quotient group ($Q$) structure
| Description: | $C_1$ |
| Order: | $1$ |
| Exponent: | $1$ |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $0$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary (for every $p$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group (for every $p$), perfect, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^4:D_6^2$, of order \(2304\)\(\medspace = 2^{8} \cdot 3^{2} \) |
| $\operatorname{Aut}(H)$ | $C_2^4:D_6^2$, of order \(2304\)\(\medspace = 2^{8} \cdot 3^{2} \) |
| $W$ | $D_6:S_4$, of order \(288\)\(\medspace = 2^{5} \cdot 3^{2} \) |
Related subgroups
| Centralizer: | $C_2$ | |||||
| Normalizer: | $C_6.\GL(2,\mathbb{Z}/4)$ | |||||
| Complements: | $C_1$ | |||||
| Maximal under-subgroups: | $A_4\times C_3:Q_8$ | $C_{12}.S_4$ | $C_{12}.S_4$ | $(C_2\times C_6):Q_{16}$ | $A_4:Q_{16}$ | $C_3^2:Q_{16}$ |
Other information
| Möbius function | $1$ |
| Projective image | $D_6:S_4$ |