Subgroup ($H$) information
| Description: | $C_6^2:D_{18}$ |
| Order: | \(1296\)\(\medspace = 2^{4} \cdot 3^{4} \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Generators: |
$a, d^{2}, b^{6}, c^{2}, b^{14}de, e^{3}, d^{3}, e^{2}$
|
| Derived length: | $3$ |
The subgroup is characteristic (hence normal), nonabelian, and monomial (hence solvable).
Ambient group ($G$) information
| Description: | $C_6^2:(D_4\times D_9)$ |
| Order: | \(5184\)\(\medspace = 2^{6} \cdot 3^{4} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Derived length: | $3$ |
The ambient group is nonabelian and monomial (hence solvable).
Quotient group ($Q$) structure
| Description: | $C_2^2$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(2\) |
| Automorphism Group: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Outer Automorphisms: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), metacyclic, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^2\times C_6^2.C_3^3.C_2^3$ |
| $\operatorname{Aut}(H)$ | $C_6^3.S_3^2$, of order \(7776\)\(\medspace = 2^{5} \cdot 3^{5} \) |
| $W$ | $C_6^2:D_{18}$, of order \(1296\)\(\medspace = 2^{4} \cdot 3^{4} \) |
Related subgroups
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $2$ |
| Projective image | $C_6^3.D_6$ |