Properties

Label 1944.2443.54.b1.a1
Order $ 2^{2} \cdot 3^{2} $
Index $ 2 \cdot 3^{3} $
Normal Yes

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Subgroup ($H$) information

Description:$C_6^2$
Order: \(36\)\(\medspace = 2^{2} \cdot 3^{2} \)
Index: \(54\)\(\medspace = 2 \cdot 3^{3} \)
Exponent: \(6\)\(\medspace = 2 \cdot 3 \)
Generators: $c^{3}, d^{3}, d^{2}, c^{2}$ Copy content Toggle raw display
Nilpotency class: $1$
Derived length: $1$

The subgroup is characteristic (hence normal), a semidirect factor, abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), and metacyclic.

Ambient group ($G$) information

Description: $C_9\times C_3^2:S_4$
Order: \(1944\)\(\medspace = 2^{3} \cdot 3^{5} \)
Exponent: \(36\)\(\medspace = 2^{2} \cdot 3^{2} \)
Derived length:$3$

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

Quotient group ($Q$) structure

Description: $S_3\times C_9$
Order: \(54\)\(\medspace = 2 \cdot 3^{3} \)
Exponent: \(18\)\(\medspace = 2 \cdot 3^{2} \)
Automorphism Group: $C_6\times S_3$, of order \(36\)\(\medspace = 2^{2} \cdot 3^{2} \)
Outer Automorphisms: $C_6$, of order \(6\)\(\medspace = 2 \cdot 3 \)
Nilpotency class: $-1$
Derived length: $2$

The quotient is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), 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)$$C_6^2:C_3.C_3^2.C_2^3$
$\operatorname{Aut}(H)$ $S_3\times \GL(2,3)$, of order \(288\)\(\medspace = 2^{5} \cdot 3^{2} \)
$\operatorname{res}(\operatorname{Aut}(G))$$D_6$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \)
$\card{\operatorname{ker}(\operatorname{res})}$\(648\)\(\medspace = 2^{3} \cdot 3^{4} \)
$W$$S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \)

Related subgroups

Centralizer:$C_3\times C_6\times C_{18}$
Normalizer:$C_9\times C_3^2:S_4$
Complements:$S_3\times C_9$ $S_3\times C_9$ $S_3\times C_9$ $S_3\times C_9$ $S_3\times C_9$ $S_3\times C_9$ $S_3\times C_9$ $S_3\times C_9$ $S_3\times C_9$
Minimal over-subgroups:$C_3\times C_6^2$$C_3^2:A_4$$C_3^2:A_4$$C_6\wr C_2$
Maximal under-subgroups:$C_3\times C_6$$C_2\times C_6$$C_2\times C_6$

Other information

Möbius function$0$
Projective image$C_9\times C_3:S_4$