Subgroup ($H$) information
| Description: | $C_{11}^2:(C_5\times Q_8)$ |
| Order: | \(4840\)\(\medspace = 2^{3} \cdot 5 \cdot 11^{2} \) |
| Index: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Exponent: | \(220\)\(\medspace = 2^{2} \cdot 5 \cdot 11 \) |
| Generators: |
$a^{5}c^{5}, c^{20}, a^{2}c^{16}, c^{110}, b^{11}c^{44}, b^{2}c^{198}$
|
| Derived length: | $2$ |
The subgroup is normal, a semidirect factor, nonabelian, supersolvable (hence solvable and monomial), and metabelian.
Ambient group ($G$) information
| Description: | $C_{55}:(Q_8\times F_{11})$ |
| Order: | \(48400\)\(\medspace = 2^{4} \cdot 5^{2} \cdot 11^{2} \) |
| Exponent: | \(220\)\(\medspace = 2^{2} \cdot 5 \cdot 11 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), and metabelian.
Quotient group ($Q$) structure
| Description: | $C_{10}$ |
| Order: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Exponent: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Automorphism Group: | $C_4$, of order \(4\)\(\medspace = 2^{2} \) |
| Outer Automorphisms: | $C_4$, of order \(4\)\(\medspace = 2^{2} \) |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,5$), hyperelementary, metacyclic, metabelian, a Z-group, 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{11}^2.C_{10}^2.C_{10}.C_2^6$ |
| $\operatorname{Aut}(H)$ | $F_{11}^2:D_4$, of order \(96800\)\(\medspace = 2^{5} \cdot 5^{2} \cdot 11^{2} \) |
| $W$ | $D_{22}:F_{11}$, of order \(4840\)\(\medspace = 2^{3} \cdot 5 \cdot 11^{2} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $10$ |
| Number of conjugacy classes in this autjugacy class | $10$ |
| Möbius function | $1$ |
| Projective image | $C_5\times D_{22}:F_{11}$ |