Subgroup ($H$) information
| Description: | $Q_8$ |
| Order: | \(8\)\(\medspace = 2^{3} \) |
| Index: | \(38880\)\(\medspace = 2^{5} \cdot 3^{5} \cdot 5 \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Generators: |
$\langle(1,68,13,72)(2,35,31,64)(3,36,16,19)(4,37,74,40)(5,27,70,52)(6,39,77,41) \!\cdots\! \rangle$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), metacyclic (hence metabelian), and rational.
Ambient group ($G$) information
| Description: | $C_3^3:S_3.C_2^4:S_5$ |
| Order: | \(311040\)\(\medspace = 2^{8} \cdot 3^{5} \cdot 5 \) |
| Exponent: | \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| Derived length: | $1$ |
The ambient group is nonabelian and nonsolvable.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^3:S_3.C_2^4:S_5$, of order \(311040\)\(\medspace = 2^{8} \cdot 3^{5} \cdot 5 \) |
| $\operatorname{Aut}(H)$ | $S_4$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| $W$ | $D_4$, of order \(8\)\(\medspace = 2^{3} \) |
Related subgroups
| Centralizer: | $D_4$ | ||||
| Normalizer: | $D_4.D_4$ | ||||
| Normal closure: | $C_3^3:S_3.C_2^4:S_5$ | ||||
| Core: | $C_1$ | ||||
| Minimal over-subgroups: | $\PSU(3,2)$ | $C_3:Q_8$ | $D_4:C_2$ | $C_2\times Q_8$ | $Q_{16}$ |
| Maximal under-subgroups: | $C_4$ | $C_4$ |
Other information
| Number of subgroups in this conjugacy class | $4860$ |
| Möbius function | $8$ |
| Projective image | $C_3^3:S_3.C_2^4:S_5$ |