Subgroup ($H$) information
| Description: | $C_2\times S_4$ |
| Order: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Index: | \(80\)\(\medspace = 2^{4} \cdot 5 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(6,7)(14,15), (6,7)(10,11), (6,14,7,15)(8,9)(12,13), (6,7)(8,9)(10,11)(12,13)(14,15), (6,10,14)(7,11,15)\rangle$
|
| Derived length: | $3$ |
The subgroup is normal, a direct factor, nonabelian, monomial (hence solvable), and rational.
Ambient group ($G$) information
| Description: | $C_2^3:F_5\times S_4$ |
| Order: | \(3840\)\(\medspace = 2^{8} \cdot 3 \cdot 5 \) |
| 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: | $D_{10}:C_4$ |
| Order: | \(80\)\(\medspace = 2^{4} \cdot 5 \) |
| Exponent: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Automorphism Group: | $D_4\times F_5$, of order \(160\)\(\medspace = 2^{5} \cdot 5 \) |
| Outer Automorphisms: | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
| Derived length: | $2$ |
The quotient is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_5\times A_4).C_2^6.C_2^5$ |
| $\operatorname{Aut}(H)$ | $C_2\times S_4$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| $\operatorname{res}(S)$ | $C_2\times S_4$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(640\)\(\medspace = 2^{7} \cdot 5 \) |
| $W$ | $S_4$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
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 | $D_{10}:C_4\times S_4$ |