Subgroup ($H$) information
| Description: | $A_5^2:C_2^2$ |
| Order: | \(14400\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 5^{2} \) |
| Index: | \(16\)\(\medspace = 2^{4} \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Generators: |
$\langle(1,8)(4,6), (11,12)(13,14), (1,3,10,2,8)(4,7,6,5,9)(11,12)(13,14)\rangle$
|
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), nonabelian, nonsolvable, and rational. Whether it is a direct factor or a semidirect factor has not been computed.
Ambient group ($G$) information
| Description: | $(C_2\times S_5^2).C_2^3$ |
| Order: | \(230400\)\(\medspace = 2^{10} \cdot 3^{2} \cdot 5^{2} \) |
| Exponent: | \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, nonsolvable, and rational.
Quotient group ($Q$) structure
| Description: | $C_2^4$ |
| Order: | \(16\)\(\medspace = 2^{4} \) |
| Exponent: | \(2\) |
| Automorphism Group: | $A_8$, of order \(20160\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 5 \cdot 7 \) |
| Outer Automorphisms: | $A_8$, of order \(20160\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 5 \cdot 7 \) |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), and rational.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^6.C_2^2.A_5^2.D_4$ |
| $\operatorname{Aut}(H)$ | $S_5^2:C_2^2$, of order \(57600\)\(\medspace = 2^{8} \cdot 3^{2} \cdot 5^{2} \) |
| $\card{W}$ | 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 |