Subgroup ($H$) information
| Description: | $C_2^3.S_5$ |
| Order: | \(960\)\(\medspace = 2^{6} \cdot 3 \cdot 5 \) |
| Index: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Generators: |
$\langle(6,8)(7,9), (1,2)(10,13,12,11), (10,12)(11,13), (6,9)(7,8)(10,12)(11,13), (1,5,4)(2,3)(6,9)(7,8)(10,11,12,13)\rangle$
|
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), a semidirect factor, nonabelian, and nonsolvable.
Ambient group ($G$) information
| Description: | $D_4\times A_4\times S_5$ |
| Order: | \(11520\)\(\medspace = 2^{8} \cdot 3^{2} \cdot 5 \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Derived length: | $2$ |
The ambient group is nonabelian and nonsolvable.
Quotient group ($Q$) structure
| Description: | $C_2\times C_6$ |
| Order: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Automorphism Group: | $D_6$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Outer Automorphisms: | $D_6$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), elementary for $p = 2$ (hence hyperelementary), and metacyclic.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(S_5\times \GL(2,\mathbb{Z}/4)).C_2^2$, of order \(46080\)\(\medspace = 2^{10} \cdot 3^{2} \cdot 5 \) |
| $\operatorname{Aut}(H)$ | $C_2^3:S_4\times S_5$, of order \(23040\)\(\medspace = 2^{9} \cdot 3^{2} \cdot 5 \) |
| $W$ | $C_6\times S_5$, of order \(720\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \) |
Related subgroups
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_2^2\times A_4\times S_5$ |