Subgroup ($H$) information
| Description: | $C_{11}^2$ |
| Order: | \(121\)\(\medspace = 11^{2} \) |
| Index: | \(440\)\(\medspace = 2^{3} \cdot 5 \cdot 11 \) |
| Exponent: | \(11\) |
| Generators: |
$c, d$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), a semidirect factor, abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), and metacyclic.
Ambient group ($G$) information
| Description: | $C_{11}^3:(C_5\times Q_8)$ |
| Order: | \(53240\)\(\medspace = 2^{3} \cdot 5 \cdot 11^{3} \) |
| 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: | $C_{44}.C_{10}$ |
| Order: | \(440\)\(\medspace = 2^{3} \cdot 5 \cdot 11 \) |
| Exponent: | \(220\)\(\medspace = 2^{2} \cdot 5 \cdot 11 \) |
| Automorphism Group: | $S_4\times F_{11}$, of order \(2640\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \cdot 11 \) |
| Outer Automorphisms: | $D_6$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian and metacyclic (hence solvable, supersolvable, monomial, and metabelian).
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{11}^3.C_{10}^2.C_5.C_2^4$ |
| $\operatorname{Aut}(H)$ | $\GL(2,11)$, of order \(13200\)\(\medspace = 2^{4} \cdot 3 \cdot 5^{2} \cdot 11 \) |
| $W$ | $C_2\times C_{10}$, of order \(20\)\(\medspace = 2^{2} \cdot 5 \) |
Related subgroups
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $0$ |
| Projective image | $C_{11}^3:(C_5\times Q_8)$ |