Subgroup ($H$) information
| Description: | $C_5\times \GL(2,3)$ |
| Order: | \(240\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| Generators: |
$a, e^{10}, c^{2}, de^{5}, e^{4}, e^{15}$
|
| Derived length: | $4$ |
The subgroup is normal, 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: | $C_2^2$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(2\) |
| Automorphism Group: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Outer Automorphisms: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), metacyclic, 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_2^2\times D_5\times A_4).C_2^5$ |
| $\operatorname{Aut}(H)$ | $C_2^4.D_6$, of order \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| $\card{W}$ | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
Related subgroups
Other information
| Möbius function | not computed |
| Projective image | not computed |