Properties

Label 972.445.18.l1.a1
Order $ 2 \cdot 3^{3} $
Index $ 2 \cdot 3^{2} $
Normal No

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Subgroup ($H$) information

Description:$C_3^2:C_6$
Order: \(54\)\(\medspace = 2 \cdot 3^{3} \)
Index: \(18\)\(\medspace = 2 \cdot 3^{2} \)
Exponent: \(6\)\(\medspace = 2 \cdot 3 \)
Generators: $\left(\begin{array}{rrrr} 2 & 0 & 2 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 2 & 0 & 1 & 1 \end{array}\right), \left(\begin{array}{rrrr} 1 & 2 & 1 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 2 & 2 & 1 \\ 0 & 1 & 2 & 0 \end{array}\right), \left(\begin{array}{rrrr} 1 & 2 & 1 & 1 \\ 2 & 2 & 0 & 2 \\ 0 & 0 & 1 & 0 \\ 2 & 1 & 0 & 0 \end{array}\right), \left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 2 & 0 & 2 & 0 \\ 1 & 0 & 0 & 2 \end{array}\right)$ Copy content Toggle raw display
Derived length: $2$

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

Ambient group ($G$) information

Description: $C_3^3.S_3^2$
Order: \(972\)\(\medspace = 2^{2} \cdot 3^{5} \)
Exponent: \(18\)\(\medspace = 2 \cdot 3^{2} \)
Derived length:$3$

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

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_3^3.S_3^2$, of order \(972\)\(\medspace = 2^{2} \cdot 3^{5} \)
$\operatorname{Aut}(H)$ $C_3^2:D_6$, of order \(108\)\(\medspace = 2^{2} \cdot 3^{3} \)
$\operatorname{res}(S)$$C_3^2:C_6$, of order \(54\)\(\medspace = 2 \cdot 3^{3} \)
$\card{\operatorname{ker}(\operatorname{res})}$\(3\)
$W$$C_3^2:C_6$, of order \(54\)\(\medspace = 2 \cdot 3^{3} \)

Related subgroups

Centralizer:$C_3$
Normalizer:$C_3^3:C_6$
Normal closure:$C_3^3:(C_3\times C_6)$
Core:$C_3^2$
Minimal over-subgroups:$C_3^3:C_6$$C_3^3:C_6$
Maximal under-subgroups:$\He_3$$C_3\times S_3$$C_3:S_3$

Other information

Number of subgroups in this conjugacy class$6$
Möbius function$0$
Projective image$C_3^3.S_3^2$