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: |
$\langle(6,7,9,10)(8,11,13,12), (1,2)(7,10)(8,13), (8,13)(11,12), (7,10)(11,12), (6,9)(7,10)(8,13)(11,12), (1,3)(2,5,4)(6,7,9,10)(8,11,13,12)\rangle$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), maximal, nonabelian, nonsolvable, and rational.
Ambient group ($G$) information
| Description: | $\OD_{16}:C_2\times S_5$ |
| Order: | \(3840\)\(\medspace = 2^{8} \cdot 3 \cdot 5 \) |
| Exponent: | \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| 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
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $D_4^2.(C_2^2\times S_5)$, of order \(30720\)\(\medspace = 2^{11} \cdot 3 \cdot 5 \) |
| $\operatorname{Aut}(H)$ | $C_2^5.C_2^3.S_5$ |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_2^4.(D_4\times S_5)$, of order \(15360\)\(\medspace = 2^{10} \cdot 3 \cdot 5 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(2\) |
| $W$ | $C_2\times C_4\times S_5$, of order \(960\)\(\medspace = 2^{6} \cdot 3 \cdot 5 \) |
Related subgroups
Other information
| Möbius function | $-1$ |
| Projective image | $C_2^2:C_4\times S_5$ |