Subgroup ($H$) information
| Description: | $C_9^4.(C_2\times C_4)$ |
| Order: | \(52488\)\(\medspace = 2^{3} \cdot 3^{8} \) |
| Index: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Generators: |
$a^{9}c^{7}d^{6}f^{7}, d^{3}, c^{12}d^{7}e^{6}f^{3}, d^{3}e^{4}, e^{3}, bd^{8}e^{3}f^{8}, c^{8}d^{6}e^{4}f^{8}, c^{6}e^{3}f^{6}, c^{12}f, f^{3}, a^{6}bc^{8}d^{8}f$
|
| Derived length: | $2$ |
The subgroup is normal, nonabelian, monomial (hence solvable), metabelian, and an A-group. 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: | $C_6$ |
| 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
While the subgroup $H$ is not characteristic, the stabilizer $S$ of $H$ in the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : S \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphisms $\operatorname{Inn}(G) \cap S$ is the Weyl group $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^6.S_3\wr D_4$, of order \(7558272\)\(\medspace = 2^{7} \cdot 3^{10} \) |
| $\operatorname{Aut}(H)$ | $C_5^4:C_2.C_2^4$, of order \(68024448\)\(\medspace = 2^{7} \cdot 3^{12} \) |
| $\card{W}$ | 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 |