Subgroup ($H$) information
| Description: | $C_2^4\times A_5$ |
| Order: | \(960\)\(\medspace = 2^{6} \cdot 3 \cdot 5 \) |
| Index: | \(3\) |
| Exponent: | \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| Generators: |
$\langle(6,7)(8,9), (6,7)(12,13), (1,2)(3,4)(10,11)(12,13), (2,5,4)(10,11), (8,9), (10,11)\rangle$
|
| Derived length: | $1$ |
The subgroup is the socle (hence characteristic and normal), maximal, a semidirect factor, nonabelian, an A-group, and nonsolvable.
Ambient group ($G$) information
| Description: | $C_2^4:\GL(2,4)$ |
| Order: | \(2880\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 5 \) |
| Exponent: | \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, an A-group, and nonsolvable.
Quotient group ($Q$) structure
| Description: | $C_3$ |
| Order: | \(3\) |
| Exponent: | \(3\) |
| 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, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group, and simple.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $S_3\times S_4\times S_5$, of order \(17280\)\(\medspace = 2^{7} \cdot 3^{3} \cdot 5 \) |
| $\operatorname{Aut}(H)$ | $S_5\times A_8$, of order \(2419200\)\(\medspace = 2^{9} \cdot 3^{3} \cdot 5^{2} \cdot 7 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $S_3^2\times S_5$, of order \(4320\)\(\medspace = 2^{5} \cdot 3^{3} \cdot 5 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(4\)\(\medspace = 2^{2} \) |
| $W$ | $\GL(2,4)$, of order \(180\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5 \) |
Related subgroups
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $A_4\times A_5$ |