Subgroup ($H$) information
| Description: | $C_2^6$ |
| Order: | \(64\)\(\medspace = 2^{6} \) |
| Index: | \(1280\)\(\medspace = 2^{8} \cdot 5 \) |
| Exponent: | \(2\) |
| Generators: |
$\langle(33,34)(35,36)(37,38)(39,40), (17,18)(19,20)(21,22)(23,24)(25,26)(27,28) \!\cdots\! \rangle$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), and rational. Whether it is a direct factor or a semidirect factor has not been computed.
Ambient group ($G$) information
| Description: | $C_2^5.C_2^8:C_{10}$ |
| Order: | \(81920\)\(\medspace = 2^{14} \cdot 5 \) |
| Exponent: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $C_2^8:C_5$ |
| Order: | \(1280\)\(\medspace = 2^{8} \cdot 5 \) |
| Exponent: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Automorphism Group: | $\AGammaL(2,16)$, of order \(62668800\)\(\medspace = 2^{14} \cdot 3^{2} \cdot 5^{2} \cdot 17 \) |
| Outer Automorphisms: | $C_3.\SL(2,16).C_4$ |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, monomial (hence solvable), metabelian, and an A-group.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $D_6^2:S_4$, of order \(10485760\)\(\medspace = 2^{21} \cdot 5 \) |
| $\operatorname{Aut}(H)$ | $\GL(6,2)$, of order \(20158709760\)\(\medspace = 2^{15} \cdot 3^{4} \cdot 5 \cdot 7^{2} \cdot 31 \) |
| $\card{W}$ | \(80\)\(\medspace = 2^{4} \cdot 5 \) |
Related subgroups
| Centralizer: | $C_2^6.C_2^4$ |
| Normalizer: | $C_2^5.C_2^8:C_{10}$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | not computed |