Subgroup ($H$) information
| Description: | $C_9:D_{18}$ |
| Order: | \(324\)\(\medspace = 2^{2} \cdot 3^{4} \) |
| Index: | \(2\) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Generators: |
$c^{9}, c^{6}, b^{3}c^{12}, a^{2}, b^{4}c^{10}, c^{14}$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), maximal, nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $C_2\times C_9^2:C_4$ |
| Order: | \(648\)\(\medspace = 2^{3} \cdot 3^{4} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Derived length: | $2$ |
The ambient group is nonabelian, monomial (hence solvable), metabelian, and an A-group.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2\times C_9^2.C_{12}.C_6.C_2$, of order \(23328\)\(\medspace = 2^{5} \cdot 3^{6} \) |
| $\operatorname{Aut}(H)$ | $C_2\times C_9^2.C_3^3.Q_8.C_3.C_6$, of order \(629856\)\(\medspace = 2^{5} \cdot 3^{9} \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_9^2:C_{24}:C_6$, of order \(11664\)\(\medspace = 2^{4} \cdot 3^{6} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(2\) |
| $W$ | $C_9^2:C_4$, of order \(324\)\(\medspace = 2^{2} \cdot 3^{4} \) |
Related subgroups
| Centralizer: | $C_2$ | ||||
| Normalizer: | $C_2\times C_9^2:C_4$ | ||||
| Minimal over-subgroups: | $C_2\times C_9^2:C_4$ | ||||
| Maximal under-subgroups: | $C_9\times C_{18}$ | $C_9:D_9$ | $C_9:D_9$ | $C_3:D_{18}$ | $C_3:D_{18}$ |
Other information
| Möbius function | $-1$ |
| Projective image | $C_9^2:C_4$ |