Subgroup ($H$) information
| Description: | $C_2^4:D_{12}$ |
| Order: | \(384\)\(\medspace = 2^{7} \cdot 3 \) |
| Index: | \(2\) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\left(\begin{array}{rr}
31 & 3 \\
0 & 1
\end{array}\right), \left(\begin{array}{rr}
41 & 0 \\
24 & 41
\end{array}\right), \left(\begin{array}{rr}
22 & 27 \\
15 & 25
\end{array}\right), \left(\begin{array}{rr}
35 & 24 \\
40 & 43
\end{array}\right), \left(\begin{array}{rr}
25 & 0 \\
0 & 25
\end{array}\right), \left(\begin{array}{rr}
25 & 0 \\
24 & 25
\end{array}\right), \left(\begin{array}{rr}
25 & 24 \\
0 & 25
\end{array}\right), \left(\begin{array}{rr}
17 & 16 \\
24 & 1
\end{array}\right)$
|
| Derived length: | $3$ |
The subgroup is normal, maximal, a direct factor, nonabelian, and monomial (hence solvable).
Ambient group ($G$) information
| Description: | $C_2^5:D_{12}$ |
| Order: | \(768\)\(\medspace = 2^{8} \cdot 3 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and monomial (hence solvable).
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_2^4\times A_4).C_2^5.C_2^4$ |
| $\operatorname{Aut}(H)$ | $A_4.C_2^6.C_2^2$ |
| $\card{W}$ | \(96\)\(\medspace = 2^{5} \cdot 3 \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $8$ |
| Number of conjugacy classes in this autjugacy class | $8$ |
| Möbius function | not computed |
| Projective image | not computed |