Subgroup ($H$) information
| Description: | $C_{11}\wr C_3$ |
| Order: | \(3993\)\(\medspace = 3 \cdot 11^{3} \) |
| Index: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Exponent: | \(33\)\(\medspace = 3 \cdot 11 \) |
| Generators: |
$\left(\begin{array}{rr}
60 & 98 \\
75 & 60
\end{array}\right), \left(\begin{array}{rr}
56 & 44 \\
33 & 67
\end{array}\right), \left(\begin{array}{rr}
78 & 0 \\
0 & 45
\end{array}\right), \left(\begin{array}{rr}
1 & 0 \\
0 & 56
\end{array}\right)$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), a semidirect factor, nonabelian, a Hall subgroup, monomial (hence solvable), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $C_5\times C_{11}\wr S_3$ |
| Order: | \(39930\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 11^{3} \) |
| Exponent: | \(330\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 11 \) |
| Derived length: | $3$ |
The ambient group is nonabelian, monomial (hence solvable), and an A-group.
Quotient group ($Q$) structure
| Description: | $C_{10}$ |
| Order: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Exponent: | \(10\)\(\medspace = 2 \cdot 5 \) |
| 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 ($p = 2,5$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{11}^2.C_{15}.C_{10}.C_2^4$ |
| $\operatorname{Aut}(H)$ | $C_{11}^2.C_{60}.C_5.C_2^3$ |
| $W$ | $C_{11}^2:S_3$, of order \(726\)\(\medspace = 2 \cdot 3 \cdot 11^{2} \) |
Related subgroups
| Centralizer: | $C_{55}$ | ||
| Normalizer: | $C_5\times C_{11}\wr S_3$ | ||
| Complements: | $C_{10}$ | ||
| Minimal over-subgroups: | $C_{11}^2:C_{165}$ | $C_{11}\wr S_3$ | |
| Maximal under-subgroups: | $C_{11}^3$ | $C_{11}^2:C_3$ | $C_{33}$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $1$ |
| Projective image | $C_5\times C_{11}^2:S_3$ |