LMFDB - Normal subgroup $C_3^2 \trianglelefteq C_3^3:D_5\wr S

Subgroup ($H$) information

Description:	$C_3^2$
Order:	$9$$\medspace = 3^{2} $
Index:	$18000$$\medspace = 2^{4} \cdot 3^{2} \cdot 5^{3} $
Exponent:	$3$
Generators:	$f^{10}, c^{2}f^{5}$ f^{10}, c^{2}f^{5}
Nilpotency class:	$1$
Derived length:	$1$

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

Ambient group ($G$) information

Description:	$C_3^3:D_5\wr S_3$
Order:	$162000$$\medspace = 2^{4} \cdot 3^{4} \cdot 5^{3} $
Exponent:	$180$$\medspace = 2^{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:	$C_5^3:(S_3\times S_4)$
Order:	$18000$$\medspace = 2^{4} \cdot 3^{2} \cdot 5^{3} $
Exponent:	$60$$\medspace = 2^{2} \cdot 3 \cdot 5 $
Automorphism Group:	$S_3\times D_5^3.D_6$, of order $72000$$\medspace = 2^{6} \cdot 3^{2} \cdot 5^{3} $
Outer Automorphisms:	$C_4$, of order $4$$\medspace = 2^{2} $
Nilpotency class:	$-1$
Derived length:	$4$

The quotient is nonabelian and monomial (hence solvable).

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 $W = G / Z_G(H)$.

$\operatorname{Aut}(G)$	$C_5^3.C_6^2.(C_{12}\times S_3^2)$
$\operatorname{Aut}(H)$	$\GL(2,3)$, of order $48$$\medspace = 2^{4} \cdot 3 $
$W$	$D_6$, of order $12$$\medspace = 2^{2} \cdot 3 $