Properties

Label 7320.a.61.a1.a1
Order $ 2^{3} \cdot 3 \cdot 5 $
Index $ 61 $
Normal No

Downloads

Learn more

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: $a^{15}, a^{40}, a^{12}, b^{61}, a^{30}$ Copy content Toggle raw display
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

Centralizer:$C_2\times C_{60}$
Normalizer:$C_2\times C_{60}$
Normal closure:$C_2\times F_{61}$
Core:$C_2$
Minimal over-subgroups:$C_2\times F_{61}$
Maximal under-subgroups:$C_2\times C_{30}$$C_{60}$$C_{60}$$C_2\times C_{20}$$C_2\times C_{12}$

Other information

Number of subgroups in this conjugacy class$61$
Möbius function$-1$
Projective image$F_{61}$