Subgroup ($H$) information
| Description: | $C_{12}:C_2^4$ |
| Order: | \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| Index: | \(2\) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\left(\begin{array}{rr}
11 & 21 \\
0 & 5
\end{array}\right), \left(\begin{array}{rr}
19 & 12 \\
0 & 19
\end{array}\right), \left(\begin{array}{rr}
7 & 12 \\
0 & 7
\end{array}\right), \left(\begin{array}{rr}
13 & 0 \\
0 & 5
\end{array}\right), \left(\begin{array}{rr}
1 & 16 \\
0 & 1
\end{array}\right), \left(\begin{array}{rr}
1 & 12 \\
0 & 1
\end{array}\right), \left(\begin{array}{rr}
1 & 3 \\
6 & 19
\end{array}\right)$
|
| Derived length: | $2$ |
The subgroup is normal, maximal, a semidirect factor, nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, metabelian, and rational.
Ambient group ($G$) information
| Description: | $S_3\times C_2^3:D_4$ |
| Order: | \(384\)\(\medspace = 2^{7} \cdot 3 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
Quotient group ($Q$) structure
| Description: | $C_2$ |
| Order: | \(2\) |
| Exponent: | \(2\) |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| 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, simple, and rational.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^8.C_3.D_6.C_2^4$ |
| $\operatorname{Aut}(H)$ | $C_2^9.(D_6\times S_4)$, of order \(147456\)\(\medspace = 2^{14} \cdot 3^{2} \) |
| $\card{W}$ | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $3$ |
| Number of conjugacy classes in this autjugacy class | $3$ |
| Möbius function | not computed |
| Projective image | not computed |