Subgroup ($H$) information
| Description: | $C_2^2\times S_5^2$ |
| Order: | \(57600\)\(\medspace = 2^{8} \cdot 3^{2} \cdot 5^{2} \) |
| Index: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Generators: |
$\langle(11,12)(13,14), (1,3)(2,4)(11,12)(13,14), (1,7,5)(2,4,10,8,6)(3,9)(11,13)(12,14), (11,13)(12,14)\rangle$
|
| 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: | $A_4\times S_5\wr C_2$ |
| Order: | \(345600\)\(\medspace = 2^{9} \cdot 3^{3} \cdot 5^{2} \) |
| Exponent: | \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| Derived length: | $2$ |
The ambient group is nonabelian and nonsolvable.
Quotient group ($Q$) structure
| Description: | $C_6$ |
| Order: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Automorphism Group: | $C_2$, of order \(2\) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,3$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $S_4\times S_5\wr C_2$, of order \(691200\)\(\medspace = 2^{10} \cdot 3^{3} \cdot 5^{2} \) |
| $\operatorname{Aut}(H)$ | $A_5^2.C_2\wr D_6$, of order \(2764800\)\(\medspace = 2^{12} \cdot 3^{3} \cdot 5^{2} \) |
| $W$ | $S_5^2:C_6$, of order \(86400\)\(\medspace = 2^{7} \cdot 3^{3} \cdot 5^{2} \) |
Related subgroups
| Centralizer: | $C_2^2$ |
| Normalizer: | $A_4\times S_5\wr C_2$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $A_4\times S_5\wr C_2$ |