Subgroup ($H$) information
Description: | $C_5\times C_3^3:A_4$ |
Order: | \(1620\)\(\medspace = 2^{2} \cdot 3^{4} \cdot 5 \) |
Index: | \(8\)\(\medspace = 2^{3} \) |
Exponent: | \(90\)\(\medspace = 2 \cdot 3^{2} \cdot 5 \) |
Generators: |
$\langle(4,9)(5,8)(10,14,13,11,12), (4,7,9), (1,3,6)(4,7,9), (1,2,4)(3,8,7)(5,9,6), (3,6)(7,9), (2,5,8), (10,11,14,12,13)\rangle$
|
Derived length: | $3$ |
The subgroup is characteristic (hence normal), a semidirect factor, nonabelian, and monomial (hence solvable).
Ambient group ($G$) information
Description: | $F_5\times S_3\wr C_3$ |
Order: | \(12960\)\(\medspace = 2^{5} \cdot 3^{4} \cdot 5 \) |
Exponent: | \(180\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5 \) |
Derived length: | $3$ |
The ambient group is nonabelian and monomial (hence solvable).
Quotient group ($Q$) structure
Description: | $C_2\times C_4$ |
Order: | \(8\)\(\medspace = 2^{3} \) |
Exponent: | \(4\)\(\medspace = 2^{2} \) |
Automorphism Group: | $D_4$, of order \(8\)\(\medspace = 2^{3} \) |
Outer Automorphisms: | $D_4$, of order \(8\)\(\medspace = 2^{3} \) |
Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), and metacyclic.
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 $W = G / Z_G(H)$.
$\operatorname{Aut}(G)$ | $F_5\times S_3\wr S_3$, of order \(25920\)\(\medspace = 2^{6} \cdot 3^{4} \cdot 5 \) |
$\operatorname{Aut}(H)$ | $C_4\times S_3\wr S_3$, of order \(5184\)\(\medspace = 2^{6} \cdot 3^{4} \) |
$W$ | $S_3^3:C_{12}$, of order \(2592\)\(\medspace = 2^{5} \cdot 3^{4} \) |
Related subgroups
Other information
Möbius function | $0$ |
Projective image | $F_5\times S_3\wr C_3$ |