Subgroup ($H$) information
| Description: | $C_2^2\times C_3^4:C_4$ |
| Order: | \(1296\)\(\medspace = 2^{4} \cdot 3^{4} \) |
| Index: | \(2\) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$d^{3}, b, ce^{2}, a^{2}, a^{4}, e^{2}, d^{2}e^{4}, e^{3}$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), maximal, nonabelian, monomial (hence solvable), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $C_6^2:F_9$ |
| Order: | \(2592\)\(\medspace = 2^{5} \cdot 3^{4} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, monomial (hence solvable), and metabelian.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^4.(C_4\times C_8).C_2^5.C_2$ |
| $\operatorname{Aut}(H)$ | $A_4.C_3^4.C_4.C_2^2.A_6.C_2^2$ |
| $\card{\operatorname{res}(\operatorname{Aut}(G))}$ | \(82944\)\(\medspace = 2^{10} \cdot 3^{4} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(2\) |
| $W$ | $C_3^2:F_9$, of order \(648\)\(\medspace = 2^{3} \cdot 3^{4} \) |
Related subgroups
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $-1$ |
| Projective image | $C_2\times C_3^2:F_9$ |