Subgroup ($H$) information
| Description: | $C_6^2:C_2^4$ |
| Order: | \(576\)\(\medspace = 2^{6} \cdot 3^{2} \) |
| Index: | \(3\) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\left(\begin{array}{rr}
31 & 24 \\
8 & 23
\end{array}\right), \left(\begin{array}{rr}
1 & 9 \\
0 & 1
\end{array}\right), \left(\begin{array}{rr}
13 & 0 \\
24 & 25
\end{array}\right), \left(\begin{array}{rr}
19 & 0 \\
0 & 19
\end{array}\right), \left(\begin{array}{rr}
35 & 0 \\
0 & 35
\end{array}\right), \left(\begin{array}{rr}
25 & 12 \\
12 & 13
\end{array}\right), \left(\begin{array}{rr}
1 & 18 \\
0 & 1
\end{array}\right), \left(\begin{array}{rr}
19 & 27 \\
0 & 1
\end{array}\right)$
|
| Derived length: | $2$ |
The subgroup is normal, maximal, a direct factor, nonabelian, supersolvable (hence solvable and monomial), and metabelian.
Ambient group ($G$) information
| Description: | $C_6^3:C_2^3$ |
| Order: | \(1728\)\(\medspace = 2^{6} \cdot 3^{3} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), and metabelian.
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
While the subgroup $H$ is not characteristic, the stabilizer $S$ of $H$ in the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : S \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphisms $\operatorname{Inn}(G) \cap S$ is the Weyl group $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^4.C_2^6.C_2^4.S_3^3.C_2$ |
| $\operatorname{Aut}(H)$ | $C_2^7.C_2^4.D_6^2$ |
| $\card{W}$ | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $4$ |
| Number of conjugacy classes in this autjugacy class | $4$ |
| Möbius function | not computed |
| Projective image | not computed |