Subgroup ($H$) information
| Description: | $C_2^6$ |
| Order: | \(64\)\(\medspace = 2^{6} \) |
| Index: | \(2304\)\(\medspace = 2^{8} \cdot 3^{2} \) |
| Exponent: | \(2\) |
| Generators: |
$\langle(5,6)(7,8)(9,10)(11,12)(17,18)(19,20)(21,22)(23,24), (9,10)(21,22), (7,8) \!\cdots\! \rangle$
|
| 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: | $C_2^{12}.(C_6\times S_3)$ |
| Order: | \(147456\)\(\medspace = 2^{14} \cdot 3^{2} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $C_2\times A_4^2:D_4$ |
| Order: | \(2304\)\(\medspace = 2^{8} \cdot 3^{2} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Automorphism Group: | $A_4^2.C_2^4.C_2^3$ |
| Outer Automorphisms: | $C_2^2\times D_4$, of order \(32\)\(\medspace = 2^{5} \) |
| 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^8.C_6^2.C_2^6.C_2^3$ |
| $\operatorname{Aut}(H)$ | $\GL(6,2)$, of order \(20158709760\)\(\medspace = 2^{15} \cdot 3^{4} \cdot 5 \cdot 7^{2} \cdot 31 \) |
| $W$ | $C_3\times S_3$, of order \(18\)\(\medspace = 2 \cdot 3^{2} \) |
Related subgroups
| Centralizer: | $C_2^{12}.C_2$ |
| Normalizer: | $C_2^{12}.(C_6\times S_3)$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | not computed |