Subgroup ($H$) information
| Description: | $C_2\times C_6^3$ |
| Order: | \(432\)\(\medspace = 2^{4} \cdot 3^{3} \) |
| Index: | \(1296\)\(\medspace = 2^{4} \cdot 3^{4} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$d^{3}f^{3}, d^{2}e^{4}g^{2}, g^{2}, e^{3}, e^{2}g^{2}, f^{3}g^{3}, g^{3}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal) and abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group). Whether it is a direct factor or a semidirect factor has not been computed.
Ambient group ($G$) information
| Description: | $C_6\wr D_6$ |
| Order: | \(559872\)\(\medspace = 2^{8} \cdot 3^{7} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Derived length: | $3$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $C_3\wr S_3\times D_4$ |
| Order: | \(1296\)\(\medspace = 2^{4} \cdot 3^{4} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Automorphism Group: | $(C_6\times \He_3).C_2^5$ |
| Outer Automorphisms: | $C_2^2\times C_6$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Nilpotency class: | $-1$ |
| Derived length: | $3$ |
The quotient is nonabelian and supersolvable (hence solvable and monomial).
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_2\times C_6^3).C_3^5.C_2^6.C_2$ |
| $\operatorname{Aut}(H)$ | $C_2\times A_8\times \SL(3,3)$ |
| $W$ | $D_6$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
Related subgroups
| Centralizer: | $C_6^6$ |
| Normalizer: | $C_6\wr D_6$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_6\wr D_6$ |