Properties

Label 6948.a.772.a1.a1
Order $ 3^{2} $
Index $ 2^{2} \cdot 193 $
Normal No

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Subgroup ($H$) information

Description:$C_3^2$
Order: \(9\)\(\medspace = 3^{2} \)
Index: \(772\)\(\medspace = 2^{2} \cdot 193 \)
Exponent: \(3\)
Generators: $a^{4}, b^{193}$ Copy content Toggle raw display
Nilpotency class: $1$
Derived length: $1$

The subgroup is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $3$-Sylow subgroup (hence a Hall subgroup), a $p$-group (hence elementary and hyperelementary), and metacyclic.

Ambient group ($G$) information

Description: $C_{579}:C_{12}$
Order: \(6948\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 193 \)
Exponent: \(2316\)\(\medspace = 2^{2} \cdot 3 \cdot 193 \)
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_{579}.C_{192}.C_2$
$\operatorname{Aut}(H)$ $\GL(2,3)$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \)
$W$$C_1$, of order $1$

Related subgroups

Centralizer:$C_3\times C_{12}$
Normalizer:$C_3\times C_{12}$
Normal closure:$C_{579}:C_3$
Core:$C_3$
Minimal over-subgroups:$C_{579}:C_3$$C_3\times C_6$
Maximal under-subgroups:$C_3$$C_3$$C_3$$C_3$

Other information

Number of subgroups in this conjugacy class$193$
Möbius function$0$
Projective image$C_{193}:C_{12}$