Subgroup ($H$) information
Description: | $C_2\times C_{60}$ |
Order: | \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
Index: | \(61\) |
Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
Generators: |
$a^{15}, a^{40}, a^{12}, b^{61}, a^{30}$
|
Nilpotency class: | $1$ |
Derived length: | $1$ |
The subgroup is maximal, abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a Hall subgroup, elementary for $p = 2$ (hence hyperelementary), and metacyclic.
Ambient group ($G$) information
Description: | $C_2\times F_{61}$ |
Order: | \(7320\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 61 \) |
Exponent: | \(3660\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 61 \) |
Derived length: | $2$ |
The ambient group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and an A-group.
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 $W = N_G(H) / Z_G(H)$.
$\operatorname{Aut}(G)$ | $C_2\times F_{61}$, of order \(7320\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 61 \) |
$\operatorname{Aut}(H)$ | $C_4^2:C_2^2$, of order \(64\)\(\medspace = 2^{6} \) |
$W$ | $C_1$, of order $1$ |
Related subgroups
Other information
Number of subgroups in this conjugacy class | $61$ |
Möbius function | $-1$ |
Projective image | $F_{61}$ |