Subgroup ($H$) information
| Description: | $C_{11}^2:F_{11}$ |
| Order: | \(13310\)\(\medspace = 2 \cdot 5 \cdot 11^{3} \) |
| Index: | $1$ |
| Exponent: | \(110\)\(\medspace = 2 \cdot 5 \cdot 11 \) |
| Generators: |
$a^{5}, b, cd^{8}, a^{2}, d$
|
| Derived length: | $3$ |
The subgroup is the radical (hence characteristic, normal, and solvable), a direct factor, nonabelian, a Hall subgroup, and supersolvable (hence monomial).
Ambient group ($G$) information
| Description: | $C_{11}^2:F_{11}$ |
| Order: | \(13310\)\(\medspace = 2 \cdot 5 \cdot 11^{3} \) |
| Exponent: | \(110\)\(\medspace = 2 \cdot 5 \cdot 11 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and supersolvable (hence solvable and monomial).
Quotient group ($Q$) structure
| Description: | $C_1$ |
| Order: | $1$ |
| Exponent: | $1$ |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $0$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary (for every $p$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group (for every $p$), perfect, and rational.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $\He_{11}.C_{10}.\PSL(2,11).C_2$, of order \(17569200\)\(\medspace = 2^{4} \cdot 3 \cdot 5^{2} \cdot 11^{4} \) |
| $\operatorname{Aut}(H)$ | $\He_{11}.C_{10}.\PSL(2,11).C_2$, of order \(17569200\)\(\medspace = 2^{4} \cdot 3 \cdot 5^{2} \cdot 11^{4} \) |
| $W$ | $C_{11}^2:F_{11}$, of order \(13310\)\(\medspace = 2 \cdot 5 \cdot 11^{3} \) |
Related subgroups
Other information
| Möbius function | $1$ |
| Projective image | $C_{11}^2:F_{11}$ |