Subgroup ($H$) information
| Description: | $C_6\times C_{18}$ |
| Order: | \(108\)\(\medspace = 2^{2} \cdot 3^{3} \) |
| Index: | \(16\)\(\medspace = 2^{4} \) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Generators: |
$b^{18}c^{6}, b^{12}, c^{6}, b^{4}, c^{4}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), and metacyclic.
Ambient group ($G$) information
| Description: | $C_2^2.(D_6\times C_{36})$ |
| Order: | \(1728\)\(\medspace = 2^{6} \cdot 3^{3} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_6.(C_2^5\times C_6).C_2^5$ |
| $\operatorname{Aut}(H)$ | $S_3\times C_3^2:D_6$, of order \(648\)\(\medspace = 2^{3} \cdot 3^{4} \) |
| $\card{W}$ | \(2\) |
Related subgroups
Other information
| Möbius function | not computed |
| Projective image | not computed |