Subgroup ($H$) information
| Description: | $S_3^4:S_3$ |
| Order: | \(7776\)\(\medspace = 2^{5} \cdot 3^{5} \) |
| Index: | $1$ |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(11,12,14), (10,15)(11,14), (1,4,3), (2,6,5), (1,2,4,6,3,5)(12,14)(13,15), (12,14), (7,9,8), (2,5)(8,9)(10,14,13,11)(12,15), (2,5)(3,4), (10,13,15)\rangle$
|
| Derived length: | $3$ |
The subgroup is the radical (hence characteristic, normal, and solvable), a direct factor, nonabelian, and a Hall subgroup. Whether it is monomial has not been computed.
Ambient group ($G$) information
| Description: | $S_3^4:S_3$ |
| Order: | \(7776\)\(\medspace = 2^{5} \cdot 3^{5} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $C_1$ |
| Order: | $1$ |
| Exponent: | $1$ |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $0$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary (for every $p$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group (for every $p$), perfect, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $S_3^5.D_4$, of order \(62208\)\(\medspace = 2^{8} \cdot 3^{5} \) |
| $\operatorname{Aut}(H)$ | $S_3^5.D_4$, of order \(62208\)\(\medspace = 2^{8} \cdot 3^{5} \) |
| $W$ | $S_3^4:S_3$, of order \(7776\)\(\medspace = 2^{5} \cdot 3^{5} \) |
Related subgroups
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $1$ |
| Projective image | $S_3^4:S_3$ |