Subgroup ($H$) information
| Description: | $C_2\times C_3^2:D_{18}$ |
| Order: | \(648\)\(\medspace = 2^{3} \cdot 3^{4} \) |
| Index: | \(2\) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Generators: |
$ad, c^{3}, c, b^{3}d, d^{4}, d^{6}, b^{2}$
|
| Derived length: | $3$ |
The subgroup is normal, maximal, a semidirect factor, nonabelian, and supersolvable (hence solvable and monomial).
Ambient group ($G$) information
| Description: | $(C_3\times C_{12}):D_{18}$ |
| Order: | \(1296\)\(\medspace = 2^{4} \cdot 3^{4} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Derived length: | $3$ |
The ambient group is nonabelian and supersolvable (hence solvable and monomial).
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
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_3\times C_9).C_6^2.C_2^4$ |
| $\operatorname{Aut}(H)$ | $C_2\times C_3^4.C_3.C_2^3$ |
| $\card{\operatorname{res}(S)}$ | \(3888\)\(\medspace = 2^{4} \cdot 3^{5} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(2\) |
| $W$ | $C_2\times C_3^2:D_{18}$, of order \(648\)\(\medspace = 2^{3} \cdot 3^{4} \) |
Related subgroups
Other information
| Möbius function | $-1$ |
| Projective image | $C_2\times C_3^2:D_{18}$ |