Subgroup ($H$) information
| Description: | $(C_{11}\times C_{44}).D_4$ |
| Order: | \(3872\)\(\medspace = 2^{5} \cdot 11^{2} \) |
| Index: | \(5\) |
| Exponent: | \(88\)\(\medspace = 2^{3} \cdot 11 \) |
| Generators: |
$a^{5}, c, d^{11}, b^{2}c, b, d^{4}, d^{22}$
|
| Derived length: | $3$ |
The subgroup is characteristic (hence normal), maximal, a semidirect factor, nonabelian, a Hall subgroup, and monomial (hence solvable).
Ambient group ($G$) information
| Description: | $C_{11}^2:(C_{10}\times \SD_{16})$ |
| Order: | \(19360\)\(\medspace = 2^{5} \cdot 5 \cdot 11^{2} \) |
| Exponent: | \(440\)\(\medspace = 2^{3} \cdot 5 \cdot 11 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and monomial (hence solvable).
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{11}^2.C_2^3.C_5.C_2^5$ |
| $\operatorname{Aut}(H)$ | $C_{11}^2.C_2^3.C_5.C_2^5$ |
| $W$ | $C_2\times D_{11}^2:C_{10}$, of order \(9680\)\(\medspace = 2^{4} \cdot 5 \cdot 11^{2} \) |
Related subgroups
Other information
| Möbius function | $-1$ |
| Projective image | $C_2\times D_{11}^2:C_{10}$ |