Subgroup ($H$) information
| Description: | not computed |
| Order: | \(8748\)\(\medspace = 2^{2} \cdot 3^{7} \) |
| Index: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | not computed |
| Generators: |
$a^{6}, gh^{2}, d^{2}eg, f^{2}h, a^{12}, c, fg^{2}h, df^{2}g^{2}h, bdef^{2}g^{2}h^{2}$
|
| Derived length: | not computed |
The subgroup is characteristic (hence normal), nonabelian, and solvable. Whether it is elementary, hyperelementary, monomial, simple, quasisimple, perfect, almost simple, or rational has not been computed.
Ambient group ($G$) information
| Description: | $C_3^6:F_9$ |
| Order: | \(52488\)\(\medspace = 2^{3} \cdot 3^{8} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $C_6$ |
| Order: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| 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 ($p = 2,3$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $D_5^3.C_2^2$, of order \(1679616\)\(\medspace = 2^{8} \cdot 3^{8} \) |
| $\operatorname{Aut}(H)$ | not computed |
| $W$ | $C_5^4.C_4^2.C_2^2$, of order \(40000\)\(\medspace = 2^{6} \cdot 5^{4} \) |
Related subgroups
| Centralizer: | $C_3^2$ | ||
| Normalizer: | $C_3^6:F_9$ | ||
| Minimal over-subgroups: | $C_3\times C_3^3.C_3^4.C_4$ | $C_3^5:F_9$ | |
| Maximal under-subgroups: | $C_3^4.C_3^3.C_2$ | $C_3^4:C_{12}$ | $(C_3^2\times \He_3):C_4$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $1$ |
| Projective image | $C_3^6:F_9$ |