Subgroup ($H$) information
| Description: | $C_4:C_{10}^2$ |
| Order: | \(400\)\(\medspace = 2^{4} \cdot 5^{2} \) |
| Index: | \(11\) |
| Exponent: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Generators: |
$a^{5}c^{4}, c^{22}, c^{33}, a^{2}c^{28}, b^{5}, b^{2}$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is maximal, nonabelian, a Hall subgroup, nilpotent (hence solvable, supersolvable, and monomial), and metabelian.
Ambient group ($G$) information
| Description: | $C_{44}:C_{10}^2$ |
| Order: | \(4400\)\(\medspace = 2^{4} \cdot 5^{2} \cdot 11 \) |
| Exponent: | \(220\)\(\medspace = 2^{2} \cdot 5 \cdot 11 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{110}.C_{10}.C_2^5$ |
| $\operatorname{Aut}(H)$ | $C_2^5.C_2^3.S_5$ |
| $\operatorname{res}(S)$ | $D_{10}.C_2^4$, of order \(320\)\(\medspace = 2^{6} \cdot 5 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(10\)\(\medspace = 2 \cdot 5 \) |
| $W$ | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $11$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $-1$ |
| Projective image | $C_2^2\times F_{11}$ |