Subgroup ($H$) information
| Description: | $(C_{11}\times C_{44}).D_4$ |
| Order: | \(3872\)\(\medspace = 2^{5} \cdot 11^{2} \) |
| Index: | \(25\)\(\medspace = 5^{2} \) |
| Exponent: | \(88\)\(\medspace = 2^{3} \cdot 11 \) |
| Generators: |
$a^{5}, cd^{60}, d^{55}, b^{2}c^{3}, b, d^{20}, d^{110}$
|
| Derived length: | $3$ |
The subgroup is characteristic (hence normal), a semidirect factor, nonabelian, a Hall subgroup, and monomial (hence solvable).
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.
Quotient group ($Q$) structure
| Description: | $C_5^2$ |
| Order: | \(25\)\(\medspace = 5^{2} \) |
| Exponent: | \(5\) |
| Automorphism Group: | $\GL(2,5)$, of order \(480\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \) |
| Outer Automorphisms: | $\GL(2,5)$, of order \(480\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \) |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), and metacyclic.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_{11}\times C_{55}).C_2^3.C_{10}.C_2^6$ |
| $\operatorname{Aut}(H)$ | $C_{11}^2.C_2^3.C_5.C_2^5$ |
| $W$ | $C_2\times D_{11}^2:C_{10}$, of order \(9680\)\(\medspace = 2^{4} \cdot 5 \cdot 11^{2} \) |
Related subgroups
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $5$ |
| Projective image | not computed |