Subgroup ($H$) information
| Description: | $C_2^2\times C_4$ |
| Order: | \(16\)\(\medspace = 2^{4} \) |
| Index: | \(663552\)\(\medspace = 2^{13} \cdot 3^{4} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Generators: |
$\langle(1,4,2,3)(5,7,6,8)(9,11,10,12)(13,15,14,16)(17,19,18,20)(21,24,22,23)(25,28,26,27) \!\cdots\! \rangle$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), and a $p$-group (hence elementary and hyperelementary). Whether it is a direct factor or a semidirect factor has not been computed.
Ambient group ($G$) information
| Description: | $D_4^3.A_4^3:A_4$ |
| Order: | \(10616832\)\(\medspace = 2^{17} \cdot 3^{4} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Derived length: | $4$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $C_2^5.A_4^3:A_4$ |
| Order: | \(663552\)\(\medspace = 2^{13} \cdot 3^{4} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Automorphism Group: | Group of order \(3822059520\)\(\medspace = 2^{20} \cdot 3^{6} \cdot 5 \) |
| Outer Automorphisms: | $C_5^3:C_{20}.Q_8$, of order \(11520\)\(\medspace = 2^{8} \cdot 3^{2} \cdot 5 \) |
| 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | Group of order \(8153726976\)\(\medspace = 2^{25} \cdot 3^{5} \) |
| $\operatorname{Aut}(H)$ | $C_2^3:S_4$, of order \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| $\card{W}$ | 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 |