Subgroup ($H$) information
| Description: | $C_{11}^2:Q_8$ |
| Order: | \(968\)\(\medspace = 2^{3} \cdot 11^{2} \) |
| Index: | \(5\) |
| Exponent: | \(44\)\(\medspace = 2^{2} \cdot 11 \) |
| Generators: |
$a^{5}, c^{4}, c^{22}, c^{11}, b$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), maximal, a semidirect factor, nonabelian, a Hall subgroup, and metacyclic (hence solvable, supersolvable, monomial, and metabelian).
Ambient group ($G$) information
| Description: | $C_{44}.F_{11}$ |
| Order: | \(4840\)\(\medspace = 2^{3} \cdot 5 \cdot 11^{2} \) |
| 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: | $C_5$ |
| Order: | \(5\) |
| Exponent: | \(5\) |
| 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, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group, and simple.
Automorphism information
Since the subgroup $H$ is characteristic, the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : \operatorname{Aut}(G) \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphism group $\operatorname{Inn}(G)$ is the Weyl group $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{11}^2.C_{10}^2.C_2^3$ |
| $\operatorname{Aut}(H)$ | $C_{110}.C_5.C_2^4$ |
| $\card{\operatorname{res}(\operatorname{Aut}(G))}$ | \(8800\)\(\medspace = 2^{5} \cdot 5^{2} \cdot 11 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(11\) |
| $W$ | $C_2\times F_{11}$, of order \(220\)\(\medspace = 2^{2} \cdot 5 \cdot 11 \) |
Related subgroups
Other information
| Möbius function | $-1$ |
| Projective image | $C_{22}:F_{11}$ |