Subgroup ($H$) information
| Description: | $A_5\times {}^2G(2,3)$ |
| Order: | \(90720\)\(\medspace = 2^{5} \cdot 3^{4} \cdot 5 \cdot 7 \) |
| Index: | $1$ |
| Exponent: | \(630\)\(\medspace = 2 \cdot 3^{2} \cdot 5 \cdot 7 \) |
| Generators: |
$\langle(1,3,8)(2,7,5)(4,9,6)(10,11)(12,13), (1,6,5)(3,4,8)(11,14,12)\rangle$
|
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), a direct factor, nonabelian, a Hall subgroup, and nonsolvable.
Ambient group ($G$) information
| Description: | $A_5\times {}^2G(2,3)$ |
| Order: | \(90720\)\(\medspace = 2^{5} \cdot 3^{4} \cdot 5 \cdot 7 \) |
| Exponent: | \(630\)\(\medspace = 2 \cdot 3^{2} \cdot 5 \cdot 7 \) |
| Derived length: | $1$ |
The ambient group is nonabelian and nonsolvable.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $S_5\times {}^2G(2,3)$, of order \(181440\)\(\medspace = 2^{6} \cdot 3^{4} \cdot 5 \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $S_5\times {}^2G(2,3)$, of order \(181440\)\(\medspace = 2^{6} \cdot 3^{4} \cdot 5 \cdot 7 \) |
| $W$ | $A_5\times {}^2G(2,3)$, of order \(90720\)\(\medspace = 2^{5} \cdot 3^{4} \cdot 5 \cdot 7 \) |
Related subgroups
Other information
| Möbius function | $1$ |
| Projective image | $A_5\times {}^2G(2,3)$ |