LMFDB - Normal subgroup $C_2^4 \trianglelefteq A_4^2:S

Subgroup ($H$) information

Description:	$C_2^4$
Order:	$16$$\medspace = 2^{4} $
Index:	$324$$\medspace = 2^{2} \cdot 3^{4} $
Exponent:	$2$
Generators:	$\langle(6,14)(8,11)(9,10)(12,13), (6,10)(8,13)(9,14)(11,12), (6,14)(9,10), (6,9)(10,14)\rangle$ (6,14)(8,11)(9,10)(12,13), (6,10)(8,13)(9,14)(11,12), (6,14)(9,10), (6,9)(10,14)
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), a $p$-group (hence elementary and hyperelementary), and rational.

Ambient group ($G$) information

Description:	$A_4^2:S_3^2$
Order:	$5184$$\medspace = 2^{6} \cdot 3^{4} $
Exponent:	$12$$\medspace = 2^{2} \cdot 3 $
Derived length:	$3$

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

Quotient group ($Q$) structure

Description:	$C_3^2:S_3^2$
Order:	$324$$\medspace = 2^{2} \cdot 3^{4} $
Exponent:	$6$$\medspace = 2 \cdot 3 $
Automorphism Group:	$C_3^2:\GL(2,3)\times D_6$, of order $5184$$\medspace = 2^{6} \cdot 3^{4} $
Outer Automorphisms:	$C_2\times S_4$, of order $48$$\medspace = 2^{4} \cdot 3 $
Nilpotency class:	$-1$
Derived length:	$2$

The quotient is nonabelian, supersolvable (hence solvable and monomial), 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 $W = G / Z_G(H)$.

$\operatorname{Aut}(G)$	$(C_3\times A_4^2).D_6^2$
$\operatorname{Aut}(H)$	$A_8$, of order $20160$$\medspace = 2^{6} \cdot 3^{2} \cdot 5 \cdot 7 $
$W$	$C_3\times S_3$, of order $18$$\medspace = 2 \cdot 3^{2} $