Subgroup ($H$) information
| Description: | $C_9^4.(C_2^2\times C_6)$ |
| Order: | \(157464\)\(\medspace = 2^{3} \cdot 3^{9} \) |
| Index: | \(2\) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Generators: |
$c^{9}, a^{6}c^{6}d^{3}e^{3}f^{3}, f^{3}, a^{4}, d^{3}e^{4}, de^{6}, d^{3}, bd^{8}e^{3}f^{8}, e^{3}, c^{14}d^{4}e^{7}f^{4}, c^{6}d^{3}e^{3}f^{3}, c^{12}e^{3}f$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), maximal, nonabelian, supersolvable (hence solvable and monomial), and metabelian. Whether it is a direct factor or a semidirect factor has not been computed.
Ambient group ($G$) information
| Description: | $C_9:D_9^3.C_6$ |
| Order: | \(314928\)\(\medspace = 2^{4} \cdot 3^{9} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Derived length: | $3$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^6.S_3\wr D_4$, of order \(7558272\)\(\medspace = 2^{7} \cdot 3^{10} \) |
| $\operatorname{Aut}(H)$ | $C_9^4.C_6\wr S_4$, of order \(204073344\)\(\medspace = 2^{7} \cdot 3^{13} \) |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | not computed |
| Autjugate subgroups: | Subgroups are not computed up to automorphism. |
Other information
| Möbius function | not computed |
| Projective image | not computed |