Properties

Label 6048.a.756.a1.a1
Order $ 2^{3} $
Index $ 2^{2} \cdot 3^{3} \cdot 7 $
Normal No

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Subgroup ($H$) information

Description:$C_8$
Order: \(8\)\(\medspace = 2^{3} \)
Index: \(756\)\(\medspace = 2^{2} \cdot 3^{3} \cdot 7 \)
Exponent: \(8\)\(\medspace = 2^{3} \)
Generators: $\left(\begin{array}{lll}\alpha & 1 & \alpha^{5} \\ \alpha^{3} & \alpha^{6} & 0 \\ 1 & \alpha^{2} & \alpha^{7} \\ \end{array}\right), \left(\begin{array}{lll}\alpha^{6} & \alpha^{2} & \alpha^{3} \\ \alpha^{7} & \alpha^{5} & 1 \\ \alpha^{7} & \alpha^{7} & \alpha^{7} \\ \end{array}\right), \left(\begin{array}{lll}\alpha^{5} & \alpha^{3} & \alpha^{4} \\ 1 & 0 & \alpha \\ 1 & 1 & \alpha^{7} \\ \end{array}\right)$ Copy content Toggle raw display
Nilpotency class: $1$
Derived length: $1$

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

Ambient group ($G$) information

Description: $\SU(3,3)$
Order: \(6048\)\(\medspace = 2^{5} \cdot 3^{3} \cdot 7 \)
Exponent: \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \)
Derived length:$0$

The ambient group is nonabelian and simple (hence nonsolvable, perfect, quasisimple, and almost simple).

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)$$G(2,2)$, of order \(12096\)\(\medspace = 2^{6} \cdot 3^{3} \cdot 7 \)
$\operatorname{Aut}(H)$ $C_2^2$, of order \(4\)\(\medspace = 2^{2} \)
$W$$C_2$, of order \(2\)

Related subgroups

Centralizer:$C_8$
Normalizer:$\OD_{16}$
Normal closure:$\SU(3,3)$
Core:$C_1$
Minimal over-subgroups:$C_3:C_8$$\OD_{16}$
Maximal under-subgroups:$C_4$

Other information

Number of subgroups in this conjugacy class$378$
Möbius function$0$
Projective image$\SU(3,3)$