Subgroup ($H$) information
| Description: | not computed |
| Order: | \(125000\)\(\medspace = 2^{3} \cdot 5^{6} \) |
| Index: | \(384\)\(\medspace = 2^{7} \cdot 3 \) |
| Exponent: | not computed |
| Generators: |
$\langle(1,3,5,2,4)(6,7,8,9,10)(11,15,14,13,12)(16,19,17,20,18), (11,15,14,13,12) \!\cdots\! \rangle$
|
| Derived length: | not computed |
The subgroup is characteristic (hence normal), nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group. Whether it is a direct factor, a semidirect factor, elementary, hyperelementary, monomial, simple, quasisimple, perfect, almost simple, or rational has not been computed.
Ambient group ($G$) information
| Description: | $C_5^6:((C_2\times C_4^3).S_4)$ |
| Order: | \(48000000\)\(\medspace = 2^{10} \cdot 3 \cdot 5^{6} \) |
| Exponent: | \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| Derived length: | $5$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $C_2^4.S_4$ |
| Order: | \(384\)\(\medspace = 2^{7} \cdot 3 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Automorphism Group: | $C_2^5.D_6^2$, of order \(4608\)\(\medspace = 2^{9} \cdot 3^{2} \) |
| Outer Automorphisms: | $S_3\times D_4$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Derived length: | $3$ |
The quotient is nonabelian and monomial (hence solvable).
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_5^6.C_2^3.C_2^4.C_6.C_4.C_2^2$ |
| $\operatorname{Aut}(H)$ | not computed |
| $\card{W}$ | \(48000000\)\(\medspace = 2^{10} \cdot 3 \cdot 5^{6} \) |
Related subgroups
| Centralizer: | $C_1$ |
| Normalizer: | $C_5^6:((C_2\times C_4^3).S_4)$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | not computed |