Subgroup ($H$) information
| Description: | $C_9\times D_{10}$ |
| Order: | \(180\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5 \) |
| Index: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | \(90\)\(\medspace = 2 \cdot 3^{2} \cdot 5 \) |
| Generators: |
$a, b^{2}, b^{6}, b^{9}, c^{6}$
|
| Derived length: | $2$ |
The subgroup is normal, a semidirect factor, nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 2$, and an A-group.
Ambient group ($G$) information
| Description: | $C_{30}.C_6^2$ |
| Order: | \(1080\)\(\medspace = 2^{3} \cdot 3^{3} \cdot 5 \) |
| Exponent: | \(180\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), and metabelian.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(D_5\times \He_3).C_2^5$ |
| $\operatorname{Aut}(H)$ | $D_{10}:C_{12}$, of order \(240\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \) |
| $\operatorname{res}(S)$ | $D_{10}:C_{12}$, of order \(240\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| $W$ | $D_{10}$, of order \(20\)\(\medspace = 2^{2} \cdot 5 \) |
Related subgroups
Other information
| Möbius function | $1$ |
| Projective image | $C_3\times D_{10}$ |