Subgroup ($H$) information
| Description: | $Q_8.D_{30}$ |
| Order: | \(480\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \) |
| Index: | \(2\) |
| Exponent: | \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| Generators: |
$ac, e^{5}, c^{2}, e^{4}, b, e^{10}, de^{15}$
|
| Derived length: | $4$ |
The subgroup is characteristic (hence normal), maximal, a semidirect factor, nonabelian, and solvable.
Ambient group ($G$) information
| Description: | $\GL(2,3):D_{10}$ |
| Order: | \(960\)\(\medspace = 2^{6} \cdot 3 \cdot 5 \) |
| Exponent: | \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| Derived length: | $4$ |
The ambient group is nonabelian and solvable.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^3\times F_5\times S_4$, of order \(3840\)\(\medspace = 2^{8} \cdot 3 \cdot 5 \) |
| $\operatorname{Aut}(H)$ | $C_2^2\times F_5\times S_4$, of order \(1920\)\(\medspace = 2^{7} \cdot 3 \cdot 5 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_2^2\times F_5\times S_4$, of order \(1920\)\(\medspace = 2^{7} \cdot 3 \cdot 5 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(2\) |
| $W$ | $D_{10}\times S_4$, of order \(480\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \) |
Related subgroups
Other information
| Möbius function | $-1$ |
| Projective image | $D_{10}\times S_4$ |