Subgroup ($H$) information
| Description: | $C_2^{12}$ |
| Order: | \(4096\)\(\medspace = 2^{12} \) |
| Index: | \(93312\)\(\medspace = 2^{7} \cdot 3^{6} \) |
| Exponent: | \(2\) |
| Generators: |
$\langle(2,9)(3,10)(5,18)(6,7)(8,16)(17,24)(19,22)(20,23), (3,10)(5,18)(6,7)(17,24) \!\cdots\! \rangle$
|
| 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, a semidirect factor, or almost simple has not been computed.
Ambient group ($G$) information
| Description: | $C_2^{12}.C_2^5:(\He_3^2:C_4)$ |
| Order: | \(382205952\)\(\medspace = 2^{19} \cdot 3^{6} \) |
| 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: | $C_2^5:(\He_3^2:C_4)$ |
| Order: | \(93312\)\(\medspace = 2^{7} \cdot 3^{6} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Automorphism Group: | $C_3^{12}.C_2^5.A_4$, of order \(373248\)\(\medspace = 2^{9} \cdot 3^{6} \) |
| Outer Automorphisms: | $C_2\times D_6$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Nilpotency class: | $-1$ |
| Derived length: | $3$ |
The quotient is nonabelian and solvable. Whether it is monomial has not been computed.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | Group of order \(4586471424\)\(\medspace = 2^{21} \cdot 3^{7} \) |
| $\operatorname{Aut}(H)$ | Group of order \(644\!\cdots\!000\)\(\medspace = 2^{66} \cdot 3^{8} \cdot 5^{3} \cdot 7^{4} \cdot 11 \cdot 13 \cdot 17 \cdot 23 \cdot 31^{2} \cdot 73 \cdot 89 \cdot 127 \) |
| $\card{W}$ | 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 |