Subgroup ($H$) information
| Description: | $C_2^9.D_4$ |
| Order: | \(4096\)\(\medspace = 2^{12} \) |
| Index: | \(2\) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Generators: |
$\langle(1,3)(5,7)(8,12)(15,18), (10,13)(11,14)(16,19)(17,20), (1,7)(3,5)(8,18) \!\cdots\! \rangle$
|
| Nilpotency class: | $4$ |
| Derived length: | $3$ |
The subgroup is normal, maximal, nonabelian, and a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary). Whether it is a direct factor or a semidirect factor has not been computed.
Ambient group ($G$) information
| Description: | $C_2^{10}.D_4$ |
| Order: | \(8192\)\(\medspace = 2^{13} \) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Nilpotency class: | $4$ |
| Derived length: | $3$ |
The ambient group is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and rational.
Quotient group ($Q$) structure
| Description: | $C_2$ |
| Order: | \(2\) |
| Exponent: | \(2\) |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Nilpotency class: | $1$ |
| 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
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | Group of order \(6442450944\)\(\medspace = 2^{31} \cdot 3 \) |
| $\operatorname{Aut}(H)$ | $C_2^{12}.C_2^6.C_6.C_2^6$ |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | not computed |
| Autjugate subgroups: | Subgroups are not computed up to automorphism. |
Other information
| Möbius function | not computed |
| Projective image | not computed |