Subgroup ($H$) information
| Description: | $C_5$ |
| Order: | \(5\) |
| Index: | \(288\)\(\medspace = 2^{5} \cdot 3^{2} \) |
| Exponent: | \(5\) |
| Generators: |
$d^{6}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), a semidirect factor, cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $5$-Sylow subgroup (hence a Hall subgroup), a $p$-group, and simple.
Ambient group ($G$) information
| Description: | $D_{10}.\SOPlus(4,2)$ |
| Order: | \(1440\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 5 \) |
| Exponent: | \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and monomial (hence solvable).
Quotient group ($Q$) structure
| Description: | $C_6^2.D_4$ |
| Order: | \(288\)\(\medspace = 2^{5} \cdot 3^{2} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Automorphism Group: | $D_6^2:D_4$, of order \(1152\)\(\medspace = 2^{7} \cdot 3^{2} \) |
| Outer Automorphisms: | $D_4$, of order \(8\)\(\medspace = 2^{3} \) |
| Nilpotency class: | $-1$ |
| Derived length: | $3$ |
The quotient is nonabelian and monomial (hence solvable).
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(D_5\times C_3:S_3).C_2^6$ |
| $\operatorname{Aut}(H)$ | $C_4$, of order \(4\)\(\medspace = 2^{2} \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_4$, of order \(4\)\(\medspace = 2^{2} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(2880\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 5 \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
| Centralizer: | $C_5\times C_3^2:\SD_{16}$ | ||||
| Normalizer: | $D_{10}.\SOPlus(4,2)$ | ||||
| Complements: | $C_6^2.D_4$ | ||||
| Minimal over-subgroups: | $C_{15}$ | $C_{15}$ | $C_{10}$ | $D_5$ | $C_{10}$ |
| Maximal under-subgroups: | $C_1$ |
Other information
| Möbius function | $0$ |
| Projective image | $D_{10}.\SOPlus(4,2)$ |