Subgroup ($H$) information
| Description: | $C_2\times D_4\times S_5$ |
| Order: | \(1920\)\(\medspace = 2^{7} \cdot 3 \cdot 5 \) |
| Index: | \(2\) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Generators: |
$\langle(6,10)(7,11)(12,15)(13,14), (6,14)(7,15)(10,12)(11,13), (8,9), (1,2)(6,12) \!\cdots\! \rangle$
|
| Derived length: | $2$ |
The subgroup is normal, maximal, a semidirect factor, nonabelian, nonsolvable, and rational.
Ambient group ($G$) information
| Description: | $(C_2^2\times D_4):S_5$ |
| Order: | \(3840\)\(\medspace = 2^{8} \cdot 3 \cdot 5 \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Derived length: | $2$ |
The ambient group is nonabelian and nonsolvable.
Quotient group ($Q$) structure
| Description: | $C_2$ |
| Order: | \(2\) |
| Exponent: | \(2\) |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group, simple, and rational.
Automorphism information
While the subgroup $H$ is not characteristic, the stabilizer $S$ of $H$ in the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : S \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphisms $\operatorname{Inn}(G) \cap S$ is the Weyl group $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_5\wr C_3:F_5$, of order \(491520\)\(\medspace = 2^{15} \cdot 3 \cdot 5 \) |
| $\operatorname{Aut}(H)$ | $C_2^5.C_2^3.S_5$ |
| $\operatorname{res}(S)$ | $C_{1348}:C_{14}$, of order \(18872\)\(\medspace = 2^{3} \cdot 7 \cdot 337 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(4\)\(\medspace = 2^{2} \) |
| $W$ | $C_2^3\times S_5$, of order \(960\)\(\medspace = 2^{6} \cdot 3 \cdot 5 \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $4$ |
| Number of conjugacy classes in this autjugacy class | $4$ |
| Möbius function | not computed |
| Projective image | $C_2^3\times S_5$ |