Subgroup ($H$) information
| Description: | $(C_2\times C_6^3):C_6$ |
| Order: | \(2592\)\(\medspace = 2^{5} \cdot 3^{4} \) |
| Index: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(2,6)(3,7), (2,7)(3,6), (3,6,7)(4,5,8)(9,10,11)(12,15,20)(13,16,17)(14,19,18) \!\cdots\! \rangle$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), a semidirect factor, nonabelian, monomial (hence solvable), and metabelian.
Ambient group ($G$) information
| Description: | $C_6^4:D_6$ |
| Order: | \(15552\)\(\medspace = 2^{6} \cdot 3^{5} \) |
| 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: | $S_3$ |
| 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_6^4.C_6^2.C_2^2$ |
| $\operatorname{Aut}(H)$ | $(C_2^3\times C_6).C_3^3.C_2^3.C_2\times S_3$ |
| $W$ | $(S_3\times C_6^2):S_4$, of order \(5184\)\(\medspace = 2^{6} \cdot 3^{4} \) |
Related subgroups
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $3$ |
| Projective image | $C_6^4:D_6$ |