Subgroup ($H$) information
| Description: | $C_2^2\times A_4\times \GL(2,5)$ |
| Order: | \(23040\)\(\medspace = 2^{9} \cdot 3^{2} \cdot 5 \) |
| Index: | $1$ |
| Exponent: | \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| Generators: |
$\left(\begin{array}{rr}
11 & 17 \\
19 & 0
\end{array}\right), \left(\begin{array}{rr}
13 & 0 \\
0 & 13
\end{array}\right), \left(\begin{array}{rr}
9 & 0 \\
0 & 9
\end{array}\right), \left(\begin{array}{rr}
17 & 4 \\
12 & 13
\end{array}\right), \left(\begin{array}{rr}
11 & 0 \\
10 & 11
\end{array}\right), \left(\begin{array}{rr}
11 & 10 \\
10 & 1
\end{array}\right), \left(\begin{array}{rr}
11 & 5 \\
15 & 16
\end{array}\right), \left(\begin{array}{rr}
11 & 0 \\
0 & 11
\end{array}\right), \left(\begin{array}{rr}
1 & 10 \\
10 & 1
\end{array}\right)$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), a direct factor, nonabelian, a Hall subgroup, and nonsolvable.
Ambient group ($G$) information
| Description: | $C_2^2\times A_4\times \GL(2,5)$ |
| Order: | \(23040\)\(\medspace = 2^{9} \cdot 3^{2} \cdot 5 \) |
| Exponent: | \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| Derived length: | $2$ |
The ambient group is nonabelian and nonsolvable.
Quotient group ($Q$) structure
| Description: | $C_1$ |
| Order: | $1$ |
| Exponent: | $1$ |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $0$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary (for every $p$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group (for every $p$), perfect, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^3.A_4^2.C_2^3.S_5$ |
| $\operatorname{Aut}(H)$ | $C_2^3.A_4^2.C_2^3.S_5$ |
| $W$ | $A_4\times S_5$, of order \(1440\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 5 \) |
Related subgroups
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $A_4\times S_5$ |