Subgroup ($H$) information
| Description: | $D_{10}$ |
| Order: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Index: | \(384\)\(\medspace = 2^{7} \cdot 3 \) |
| Exponent: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Generators: |
$\langle(1,5,2,3,4), (6,7)(8,9), (2,3)(4,5)(8,9)\rangle$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), a semidirect factor, nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 2$, and an A-group.
Ambient group ($G$) information
| Description: | $C_2\wr S_3\times F_5$ |
| Order: | \(7680\)\(\medspace = 2^{9} \cdot 3 \cdot 5 \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $C_2^5.D_6$ |
| Order: | \(384\)\(\medspace = 2^{7} \cdot 3 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Automorphism Group: | $C_2^4:D_6^2$, of order \(2304\)\(\medspace = 2^{8} \cdot 3^{2} \) |
| Outer Automorphisms: | $C_2\times D_6$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_5^4.\OD_{16}$, of order \(92160\)\(\medspace = 2^{11} \cdot 3^{2} \cdot 5 \) |
| $\operatorname{Aut}(H)$ | $C_2\times F_5$, of order \(40\)\(\medspace = 2^{3} \cdot 5 \) |
| $W$ | $C_2\times F_5$, of order \(40\)\(\medspace = 2^{3} \cdot 5 \) |
Related subgroups
| Centralizer: | $C_2^2\wr C_3$ |
| Normalizer: | $C_2\wr S_3\times F_5$ |
| Complements: | $C_2^5.D_6$ |
| Minimal over-subgroups: | $C_2\times D_{10}$ |
| Maximal under-subgroups: | $C_{10}$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_2^3:S_4\times F_5$ |