Subgroup ($H$) information
| Description: | $Q_{16}\times C_{22}$ |
| Order: | \(352\)\(\medspace = 2^{5} \cdot 11 \) |
| Index: | \(5\) |
| Exponent: | \(88\)\(\medspace = 2^{3} \cdot 11 \) |
| Generators: |
$a, c^{44}, c^{11}, c^{8}, b^{5}, c^{66}$
|
| Nilpotency class: | $3$ |
| Derived length: | $2$ |
The subgroup is the Fitting subgroup (hence characteristic, normal, nilpotent, solvable, supersolvable, and monomial), maximal, a semidirect factor, nonabelian, a Hall subgroup, elementary for $p = 2$ (hence hyperelementary), and metabelian.
Ambient group ($G$) information
| Description: | $C_{11}:C_{10}\times Q_{16}$ |
| Order: | \(1760\)\(\medspace = 2^{5} \cdot 5 \cdot 11 \) |
| Exponent: | \(440\)\(\medspace = 2^{3} \cdot 5 \cdot 11 \) |
| 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 \(4\)\(\medspace = 2^{2} \) |
| Nilpotency class: | $1$ |
| 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_2\times C_{44}).C_{10}.C_2^5$ |
| $\operatorname{Aut}(H)$ | $C_{10}\times C_2^3.C_2^4.C_2$ |
| $\card{\operatorname{res}(\operatorname{Aut}(G))}$ | \(2560\)\(\medspace = 2^{9} \cdot 5 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(11\) |
| $W$ | $C_5\times D_4$, of order \(40\)\(\medspace = 2^{3} \cdot 5 \) |
Related subgroups
Other information
| Möbius function | $-1$ |
| Projective image | $C_{44}:C_{10}$ |