Subgroup ($H$) information
| Description: | $C_1$ |
| Order: | $1$ |
| Index: | \(1458\)\(\medspace = 2 \cdot 3^{6} \) |
| Exponent: | $1$ |
| Generators: | |
| Nilpotency class: | $0$ |
| Derived length: | $0$ |
The subgroup is the center (hence characteristic, normal, abelian, central, nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a direct factor, cyclic (hence elementary (for every $p$), hyperelementary, metacyclic, and a Z-group), stem, a $p$-group (for every $p$), perfect, and rational.
Ambient group ($G$) information
| Description: | $(C_3\times C_9^2):C_6$ |
| Order: | \(1458\)\(\medspace = 2 \cdot 3^{6} \) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), and metabelian.
Quotient group ($Q$) structure
| Description: | $(C_3\times C_9^2):C_6$ |
| Order: | \(1458\)\(\medspace = 2 \cdot 3^{6} \) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Automorphism Group: | $C_9^2.C_3^5.C_2^2$, of order \(78732\)\(\medspace = 2^{2} \cdot 3^{9} \) |
| Outer Automorphisms: | $S_3\times C_3^2$, of order \(54\)\(\medspace = 2 \cdot 3^{3} \) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, supersolvable (hence solvable and monomial), and metabelian.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_9^2.C_3^5.C_2^2$, of order \(78732\)\(\medspace = 2^{2} \cdot 3^{9} \) |
| $\operatorname{Aut}(H)$ | $C_1$, of order $1$ |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $(C_3\times C_9^2):C_6$ | |||||||
| Normalizer: | $(C_3\times C_9^2):C_6$ | |||||||
| Complements: | $(C_3\times C_9^2):C_6$ | |||||||
| Minimal over-subgroups: | $C_3$ | $C_3$ | $C_3$ | $C_3$ | $C_3$ | $C_3$ | $C_3$ | $C_2$ |
Other information
| Möbius function | $0$ |
| Projective image | $(C_3\times C_9^2):C_6$ |