Properties

Label 239148450.h.450._.A
Order $ 3^{12} $
Index $ 2 \cdot 3^{2} \cdot 5^{2} $
Normal Yes

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Subgroup ($H$) information

Description:not computed
Order: \(531441\)\(\medspace = 3^{12} \)
Index: \(450\)\(\medspace = 2 \cdot 3^{2} \cdot 5^{2} \)
Exponent: not computed
Generators: $\langle(16,18,17)(22,23,24)(25,27,26)(34,36,35)(40,42,41), (34,35,36)(43,45,44) \!\cdots\! \rangle$ Copy content Toggle raw display
Nilpotency class: not computed
Derived length: not computed

The subgroup is characteristic (hence normal), abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), and a $p$-group (hence elementary and hyperelementary). Whether it is a direct factor, a semidirect factor, elementary, hyperelementary, monomial, simple, quasisimple, perfect, almost simple, or rational has not been computed.

Ambient group ($G$) information

Description: $C_3^{12}.C_5^2.C_3.C_6$
Order: \(239148450\)\(\medspace = 2 \cdot 3^{14} \cdot 5^{2} \)
Exponent: \(90\)\(\medspace = 2 \cdot 3^{2} \cdot 5 \)
Derived length:$4$

The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.

Quotient group ($Q$) structure

Description: $C_3\times C_5^2:S_3$
Order: \(450\)\(\medspace = 2 \cdot 3^{2} \cdot 5^{2} \)
Exponent: \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \)
Automorphism Group: $C_5^2:(C_4\times D_6)$, of order \(1200\)\(\medspace = 2^{4} \cdot 3 \cdot 5^{2} \)
Outer Automorphisms: $C_2\times C_4$, of order \(8\)\(\medspace = 2^{3} \)
Nilpotency class: $-1$
Derived length: $3$

The quotient is nonabelian, monomial (hence solvable), and an A-group.

Automorphism information

Since the subgroup $H$ is characteristic, the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : \operatorname{Aut}(G) \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphism group $\operatorname{Inn}(G)$ is the Weyl group $W = G / Z_G(H)$.

$\operatorname{Aut}(G)$Group of order \(5101833600\)\(\medspace = 2^{7} \cdot 3^{13} \cdot 5^{2} \)
$\operatorname{Aut}(H)$ not computed
$\card{W}$ not computed

Related subgroups

Centralizer: not computed
Normalizer: not computed
Autjugate subgroups: Subgroups are not computed up to automorphism.

Other information

Möbius function not computed
Projective image not computed