Subgroup ($H$) information
| Description: | $C_3\times C_{36}$ |
| Order: | \(108\)\(\medspace = 2^{2} \cdot 3^{3} \) |
| Index: | \(16\)\(\medspace = 2^{4} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Generators: |
$c^{12}, b^{6}, b^{14}, c^{16}, c^{24}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), elementary for $p = 3$ (hence hyperelementary), and metacyclic.
Ambient group ($G$) information
| Description: | $C_{48}.D_{18}$ |
| Order: | \(1728\)\(\medspace = 2^{6} \cdot 3^{3} \) |
| Exponent: | \(144\)\(\medspace = 2^{4} \cdot 3^{2} \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), and metabelian.
Quotient group ($Q$) structure
| Description: | $C_2^2\times C_4$ |
| Order: | \(16\)\(\medspace = 2^{4} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Automorphism Group: | $C_2^3:S_4$, of order \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| Outer Automorphisms: | $C_2^3:S_4$, of order \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group) and a $p$-group (hence elementary and hyperelementary).
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{18}.(C_2^4\times C_{12}).C_2$ |
| $\operatorname{Aut}(H)$ | $C_6^2:S_3$, of order \(216\)\(\medspace = 2^{3} \cdot 3^{3} \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_2^2\times C_6$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(288\)\(\medspace = 2^{5} \cdot 3^{2} \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
Other information
| Möbius function | $0$ |
| Projective image | $C_4\times D_{18}$ |