Subgroup ($H$) information
| Description: | $S_5^2:C_2^2$ |
| Order: | \(57600\)\(\medspace = 2^{8} \cdot 3^{2} \cdot 5^{2} \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| Generators: |
$\langle(11,12)(13,14), (1,2)(13,14), (1,9,3,5,8,4)(2,7,10,6)(11,13,12,14)\rangle$
|
| Derived length: | $2$ |
The subgroup is 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: | $C_2^2$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(2\) |
| Automorphism Group: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Outer Automorphisms: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), metacyclic, and rational.
Automorphism information
While the subgroup $H$ is not characteristic, the stabilizer $S$ of $H$ in the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : S \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphisms $\operatorname{Inn}(G) \cap S$ is the Weyl group $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^6.C_2^2.A_5^2.D_4$ |
| $\operatorname{Aut}(H)$ | $C_2^2\times S_5\wr C_2$, of order \(115200\)\(\medspace = 2^{9} \cdot 3^{2} \cdot 5^{2} \) |
| $\card{W}$ | 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 |