Subgroup ($H$) information
| Description: | $C_5\times C_3^3:A_4$ |
| Order: | \(1620\)\(\medspace = 2^{2} \cdot 3^{4} \cdot 5 \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(90\)\(\medspace = 2 \cdot 3^{2} \cdot 5 \) |
| Generators: |
$\langle(4,9)(6,8), (10,14,12,13,11), (4,9)(5,7), (3,9,4), (1,4,7,6,3,2,8,9,5)(10,12,11,14,13), (1,6,8)(3,4,9), (2,7,5)\rangle$
|
| Derived length: | $3$ |
The subgroup is the commutator subgroup (hence characteristic and normal), a semidirect factor, nonabelian, and monomial (hence solvable).
Ambient group ($G$) information
| Description: | $D_5\times C_3^3:S_4$ |
| Order: | \(6480\)\(\medspace = 2^{4} \cdot 3^{4} \cdot 5 \) |
| Exponent: | \(180\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5 \) |
| Derived length: | $4$ |
The ambient group is nonabelian and monomial (hence solvable).
Quotient group ($Q$) structure
| Description: | $C_2^2$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(2\) |
| Automorphism Group: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Outer Automorphisms: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), metacyclic, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $F_5\times S_3\wr S_3$, of order \(25920\)\(\medspace = 2^{6} \cdot 3^{4} \cdot 5 \) |
| $\operatorname{Aut}(H)$ | $C_4\times S_3\wr S_3$, of order \(5184\)\(\medspace = 2^{6} \cdot 3^{4} \) |
| $W$ | $C_2\times C_3^3:S_4$, of order \(1296\)\(\medspace = 2^{4} \cdot 3^{4} \) |
Related subgroups
Other information
| Möbius function | $2$ |
| Projective image | $D_5\times C_3^3:S_4$ |