Subgroup ($H$) information
| Description: | $C_2^{10}$ |
| Order: | \(1024\)\(\medspace = 2^{10} \) |
| Index: | \(2304\)\(\medspace = 2^{8} \cdot 3^{2} \) |
| Exponent: | \(2\) |
| Generators: |
$\langle(19,20)(21,22), (7,8)(9,10)(19,20)(21,22), (3,5)(4,6)(11,14)(12,13)(15,16) \!\cdots\! \rangle$
|
| 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: | $C_2^{12}.A_4^2:C_2^2$ |
| Order: | \(2359296\)\(\medspace = 2^{18} \cdot 3^{2} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Derived length: | $4$ |
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^9.A_4^2.C_2^6.C_2^5$ |
| $\operatorname{Aut}(H)$ | Group of order \(366\!\cdots\!200\)\(\medspace = 2^{45} \cdot 3^{6} \cdot 5^{2} \cdot 7^{3} \cdot 11 \cdot 17 \cdot 31^{2} \cdot 73 \cdot 127 \) |
| $\card{W}$ | 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 |