Subgroup ($H$) information
| Description: | $(C_3\times C_9^2):C_6$ |
| Order: | \(1458\)\(\medspace = 2 \cdot 3^{6} \) |
| Index: | \(2\) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Generators: |
$b^{3}, d^{3}, e^{7}, b^{2}d^{6}e^{6}, e^{3}, ce^{5}, d^{7}e^{6}$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), maximal, a semidirect factor, nonabelian, supersolvable (hence solvable and monomial), and metabelian.
Ambient group ($G$) information
| Description: | $C_9^2.S_3^2$ |
| Order: | \(2916\)\(\medspace = 2^{2} \cdot 3^{6} \) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| 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
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_3\times C_9).C_3^5.C_2^2$, of order \(26244\)\(\medspace = 2^{2} \cdot 3^{8} \) |
| $\operatorname{Aut}(H)$ | $C_9^2.C_3^5.C_2^2$, of order \(78732\)\(\medspace = 2^{2} \cdot 3^{9} \) |
| $\card{\operatorname{res}(\operatorname{Aut}(G))}$ | \(26244\)\(\medspace = 2^{2} \cdot 3^{8} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | $1$ |
| $W$ | $C_9^2.S_3^2$, of order \(2916\)\(\medspace = 2^{2} \cdot 3^{6} \) |
Related subgroups
| Centralizer: | $C_1$ | ||||
| Normalizer: | $C_9^2.S_3^2$ | ||||
| Complements: | $C_2$ $C_2$ | ||||
| Minimal over-subgroups: | $C_9^2.S_3^2$ | ||||
| Maximal under-subgroups: | $C_9^2.C_3^2$ | $C_9^2:S_3$ | $(C_3^2\times C_9):C_6$ | $C_9^2:C_6$ | $C_9^2:C_6$ |
Other information
| Möbius function | $-1$ |
| Projective image | $C_9^2.S_3^2$ |