Subgroup ($H$) information
| Description: | $C_8:C_{10}^2$ |
| Order: | \(800\)\(\medspace = 2^{5} \cdot 5^{2} \) |
| Index: | \(121\)\(\medspace = 11^{2} \) |
| Exponent: | \(40\)\(\medspace = 2^{3} \cdot 5 \) |
| Generators: |
$a^{5}, b^{2}c^{3}, d^{55}, a^{2}d^{60}, bc^{3}d^{180}, d^{110}, d^{44}$
|
| Nilpotency class: | $3$ |
| Derived length: | $2$ |
The subgroup is maximal, nonabelian, a Hall subgroup, nilpotent (hence solvable, supersolvable, and monomial), and metabelian.
Ambient group ($G$) information
| Description: | $C_{11}^2:C_8:C_{10}^2$ |
| Order: | \(96800\)\(\medspace = 2^{5} \cdot 5^{2} \cdot 11^{2} \) |
| Exponent: | \(440\)\(\medspace = 2^{3} \cdot 5 \cdot 11 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_{11}\times C_{55}).C_2^3.C_{10}.C_2^6$ |
| $\operatorname{Aut}(H)$ | $C_2^5.C_2^4.S_5$ |
| $W$ | $D_4$, of order \(8\)\(\medspace = 2^{3} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $121$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $-1$ |
| Projective image | $C_2\times D_{11}^2:C_{10}$ |