Subgroup ($H$) information
| Description: | $C_4\times A_5$ |
| Order: | \(240\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \) |
| Index: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Generators: |
$\langle(5,8,6,7)(9,10)(11,12), (5,8,6,7)(9,13,11), (5,7,6,8), (5,6)(7,8)\rangle$
|
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), a semidirect factor, nonabelian, an A-group, and nonsolvable.
Ambient group ($G$) information
| Description: | $D_4\times A_4\times A_5$ |
| Order: | \(5760\)\(\medspace = 2^{7} \cdot 3^{2} \cdot 5 \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Derived length: | $2$ |
The ambient group is nonabelian and nonsolvable.
Quotient group ($Q$) structure
| Description: | $C_2\times A_4$ |
| Order: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Automorphism Group: | $S_4$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Derived length: | $2$ |
The quotient is nonabelian, monomial (hence solvable), metabelian, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $D_4\times S_4\times S_5$, of order \(23040\)\(\medspace = 2^{9} \cdot 3^{2} \cdot 5 \) |
| $\operatorname{Aut}(H)$ | $C_2\times S_5$, of order \(240\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \) |
| $W$ | $C_2\times A_5$, of order \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
Related subgroups
Other information
| Möbius function | $-4$ |
| Projective image | $C_2^4:\GL(2,4)$ |