Subgroup ($H$) information
| Description: | $C_2^2\wr C_2$ |
| Order: | \(32\)\(\medspace = 2^{5} \) |
| Index: | \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Generators: |
$\left(\begin{array}{rr}
43 & 30 \\
0 & 1
\end{array}\right), \left(\begin{array}{rr}
21 & 0 \\
22 & 21
\end{array}\right), \left(\begin{array}{rr}
8 & 17 \\
35 & 14
\end{array}\right)$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), metabelian, and rational.
Ambient group ($G$) information
| Description: | $C_2^2:D_{12}\times C_{10}$ |
| Order: | \(960\)\(\medspace = 2^{6} \cdot 3 \cdot 5 \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3:(C_2^6.C_2^5.C_2^4)$ |
| $\operatorname{Aut}(H)$ | $C_2\wr S_3$, of order \(384\)\(\medspace = 2^{7} \cdot 3 \) |
| $\operatorname{res}(S)$ | $C_2^3\wr C_2$, of order \(128\)\(\medspace = 2^{7} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(32\)\(\medspace = 2^{5} \) |
| $W$ | $C_2^3$, of order \(8\)\(\medspace = 2^{3} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $24$ |
| Number of conjugacy classes in this autjugacy class | $8$ |
| Möbius function | $-1$ |
| Projective image | $C_{15}:C_2^4$ |