Subgroup ($H$) information
| Description: | $C_6^4.(C_3\times S_3)$ |
| Order: | \(23328\)\(\medspace = 2^{5} \cdot 3^{6} \) |
| Index: | \(2\) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Generators: |
$a^{3}, d^{3}e^{3}f, c^{4}d^{4}e^{2}, b^{6}, e^{3}, a^{2}, b^{8}c^{3}d^{3}f, c^{3}d^{3}, c^{2}d^{2}, f, d^{2}e^{2}$
|
| Derived length: | $3$ |
The subgroup is characteristic (hence normal), maximal, a semidirect factor, nonabelian, and solvable. Whether it is monomial has not been computed.
Ambient group ($G$) information
| Description: | $C_6^4.S_3^2$ |
| Order: | \(46656\)\(\medspace = 2^{6} \cdot 3^{6} \) |
| 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: | $C_2$ |
| 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_2\times C_6^3).C_3^4.C_2^3$ |
| $\operatorname{Aut}(H)$ | $C_6^4.C_3^5.C_2^3$ |
| $W$ | $C_6^4.D_6$, of order \(15552\)\(\medspace = 2^{6} \cdot 3^{5} \) |
Related subgroups
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_6^4.S_3^2$ |