Subgroup ($H$) information
| Description: | $C_2^{12}$ |
| Order: | \(4096\)\(\medspace = 2^{12} \) |
| Index: | \(559872\)\(\medspace = 2^{8} \cdot 3^{7} \) |
| Exponent: | \(2\) |
| Generators: |
$\langle(1,6)(4,23)(5,17)(7,21)(10,15)(11,19)(14,16)(20,22), (3,13)(8,24), (1,20) \!\cdots\! \rangle$
|
| 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: | $C_2^{12}.(C_3^6.C_2^5:S_4)$ |
| Order: | \(2293235712\)\(\medspace = 2^{20} \cdot 3^{7} \) |
| Exponent: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Derived length: | $5$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $C_3^6.C_2^5:S_4$ |
| Order: | \(559872\)\(\medspace = 2^{8} \cdot 3^{7} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Automorphism Group: | $C_3^6.C_2^7:S_4$, of order \(2239488\)\(\medspace = 2^{10} \cdot 3^{7} \) |
| Outer Automorphisms: | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
| Nilpotency class: | $-1$ |
| Derived length: | $4$ |
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^{12}.(C_3^6.C_2^7:S_4)$, of order \(9172942848\)\(\medspace = 2^{22} \cdot 3^{7} \) |
| $\operatorname{Aut}(H)$ | 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 |