Subgroup ($H$) information
| Description: | $C_{10}.D_{10}$ |
| Order: | \(200\)\(\medspace = 2^{3} \cdot 5^{2} \) |
| Index: | \(14641\)\(\medspace = 11^{4} \) |
| Exponent: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Generators: |
$ac^{2}d^{9}e^{5}f^{6}g^{5}, a^{2}c^{4}d^{48}e^{9}fg^{7}, cd^{33}e^{3}f^{5}g^{5}, be^{3}f^{5}, d^{11}e^{4}$
|
| Derived length: | $2$ |
The subgroup is maximal, nonabelian, a Hall subgroup, supersolvable (hence solvable and monomial), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $C_{11}^4:(C_{10}.D_{10})$ |
| Order: | \(2928200\)\(\medspace = 2^{3} \cdot 5^{2} \cdot 11^{4} \) |
| Exponent: | \(220\)\(\medspace = 2^{2} \cdot 5 \cdot 11 \) |
| Derived length: | $3$ |
The ambient group is nonabelian, monomial (hence solvable), and an A-group.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{11}^4.C_5^2.C_2^3.C_5.C_2^3$ |
| $\operatorname{Aut}(H)$ | $F_5^2:D_4$, of order \(3200\)\(\medspace = 2^{7} \cdot 5^{2} \) |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | not computed |
| Normal closure: | not computed |
| Core: | not computed |
| Autjugate subgroups: | Subgroups are not computed up to automorphism. |
Other information
| Number of subgroups in this conjugacy class | $14641$ |
| Möbius function | not computed |
| Projective image | not computed |