Subgroup ($H$) information
| Description: | $C_{11}^2$ |
| Order: | \(121\)\(\medspace = 11^{2} \) |
| Index: | \(33\)\(\medspace = 3 \cdot 11 \) |
| Exponent: | \(11\) |
| Generators: |
$a^{3}b^{4}c^{6}, c$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is 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: | $C_{11}\wr C_3$ |
| Order: | \(3993\)\(\medspace = 3 \cdot 11^{3} \) |
| Exponent: | \(33\)\(\medspace = 3 \cdot 11 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, monomial (hence solvable), metabelian, and an A-group.
Quotient set structure
Since this subgroup has trivial core, the ambient group $G$ acts faithfully and transitively on the set of cosets of $H$. The resulting permutation representation is isomorphic to 33T23.
Automorphism information
While the subgroup $H$ is not characteristic, the stabilizer $S$ of $H$ in the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : S \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphisms $\operatorname{Inn}(G) \cap S$ is the Weyl group $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{11}^2.C_{60}.C_5.C_2^3$ |
| $\operatorname{Aut}(H)$ | $\GL(2,11)$, of order \(13200\)\(\medspace = 2^{4} \cdot 3 \cdot 5^{2} \cdot 11 \) |
| $\operatorname{res}(S)$ | $C_2\times C_{10}$, of order \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(121\)\(\medspace = 11^{2} \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $3$ |
| Möbius function | $0$ |
| Projective image | $C_{11}\wr C_3$ |