Subgroup ($H$) information
| Description: | $C_2^4.D_6\times A_5$ |
| Order: | \(11520\)\(\medspace = 2^{8} \cdot 3^{2} \cdot 5 \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Generators: |
$\langle(6,7)(8,9), (2,4,5)(6,7)(10,11)(12,14,13,15), (8,10,9,11)(12,13)(14,15) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is characteristic (hence normal), nonabelian, and nonsolvable. Whether it is a direct factor or a semidirect factor has not been computed.
Ambient group ($G$) information
| Description: | $(S_5\times \GL(2,\mathbb{Z}/4)).C_2^2$ |
| Order: | \(46080\)\(\medspace = 2^{10} \cdot 3^{2} \cdot 5 \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Derived length: | $3$ |
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
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{11}^2:C_{44}$, of order \(2949120\)\(\medspace = 2^{16} \cdot 3^{2} \cdot 5 \) |
| $\operatorname{Aut}(H)$ | $(C_2^3\times A_4).C_2^3.S_5$ |
| $W$ | $C_2\times S_4\times S_5$, of order \(5760\)\(\medspace = 2^{7} \cdot 3^{2} \cdot 5 \) |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $(S_5\times \GL(2,\mathbb{Z}/4)).C_2^2$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_2^2\times S_4\times S_5$ |