Subgroup ($H$) information
| Description: | $C_6^4.S_3^2$ |
| Order: | \(46656\)\(\medspace = 2^{6} \cdot 3^{6} \) |
| Index: | \(2\) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Generators: |
$ae^{4}f^{2}, b^{3}f^{4}g, f^{3}, c^{2}, e^{2}g^{2}, d^{4}, d^{6}e^{3}f^{3}, b^{2}c^{3}d^{9}e^{4}f^{5}, e^{3}f^{3}, e^{4}f^{2}g, c^{3}d^{9}f^{3}, f^{2}g^{2}$
|
| Derived length: | $3$ |
The subgroup is characteristic (hence normal), maximal, nonabelian, and solvable. Whether it is a direct factor, a semidirect factor, or monomial has not been computed.
Ambient group ($G$) information
| Description: | $C_6^4.(S_3\times D_6)$ |
| Order: | \(93312\)\(\medspace = 2^{7} \cdot 3^{6} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Derived length: | $3$ |
The ambient group is nonabelian, solvable, and rational. Whether it is monomial has not been computed.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^{16}.C_5^4.\SD_{16}$, of order \(559872\)\(\medspace = 2^{8} \cdot 3^{7} \) |
| $\operatorname{Aut}(H)$ | $C_5^3:C_{20}:D_4$, of order \(559872\)\(\medspace = 2^{8} \cdot 3^{7} \) |
| $W$ | $C_6^4.(S_3\times D_6)$, of order \(93312\)\(\medspace = 2^{7} \cdot 3^{6} \) |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $C_6^4.(S_3\times D_6)$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_6^4.(S_3\times D_6)$ |