Properties

Label 1458.1107.54.bg1
Order $ 3^{3} $
Index $ 2 \cdot 3^{3} $
Normal No

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Subgroup ($H$) information

Description:$C_3^3$
Order: \(27\)\(\medspace = 3^{3} \)
Index: \(54\)\(\medspace = 2 \cdot 3^{3} \)
Exponent: \(3\)
Generators: $a, d, e$ 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) and a $p$-group (hence elementary and hyperelementary).

Ambient group ($G$) information

Description: $C_3^5:C_6$
Order: \(1458\)\(\medspace = 2 \cdot 3^{6} \)
Exponent: \(18\)\(\medspace = 2 \cdot 3^{2} \)
Derived length:$2$

The ambient group is nonabelian, supersolvable (hence solvable and monomial), and metabelian.

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_3^4.S_3^3$, of order \(17496\)\(\medspace = 2^{3} \cdot 3^{7} \)
$\operatorname{Aut}(H)$ $\GL(3,3)$, of order \(11232\)\(\medspace = 2^{5} \cdot 3^{3} \cdot 13 \)
$\operatorname{res}(S)$$C_2\times D_6$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \)
$\card{\operatorname{ker}(\operatorname{res})}$\(81\)\(\medspace = 3^{4} \)
$W$$C_2$, of order \(2\)

Related subgroups

Centralizer:$C_3^5$
Normalizer:$C_3^4:C_6$
Normal closure:$C_3^5$
Core:$C_3^2$
Minimal over-subgroups:$C_3^4$$C_3^4$$C_3^2:C_6$
Maximal under-subgroups:$C_3^2$$C_3^2$$C_3^2$$C_3^2$$C_3^2$

Other information

Number of subgroups in this autjugacy class$9$
Number of conjugacy classes in this autjugacy class$3$
Möbius function$0$
Projective image$C_3^5:C_6$