Subgroup ($H$) information
| Description: | not computed |
| Order: | \(93312\)\(\medspace = 2^{7} \cdot 3^{6} \) |
| Index: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | not computed |
| Generators: |
$b^{3}, c^{3}, d^{2}e^{4}g^{2}, g^{3}, f^{2}g^{2}, b^{2}f^{2}g^{2}, f^{3}g^{3}, e^{3}, e^{2}g^{2}, d^{4}e^{2}, a^{2}, d^{3}f^{3}, c^{6}d^{3}f^{3}$
|
| Derived length: | not computed |
The subgroup is characteristic (hence normal), nonabelian, supersolvable (hence solvable and monomial), and metabelian. Whether it is a direct factor, a semidirect factor, elementary, hyperelementary, monomial, simple, quasisimple, perfect, almost simple, or rational 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: | $S_3$ |
| Order: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Automorphism Group: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $2$ |
The quotient is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), hyperelementary for $p = 2$, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_2\times C_6^3).C_3^5.C_2^6.C_2$ |
| $\operatorname{Aut}(H)$ | not computed |
| $W$ | $C_6^3:D_6$, of order \(2592\)\(\medspace = 2^{5} \cdot 3^{4} \) |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $C_6\wr D_6$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_2\times C_6^4.S_3^2$ |