LMFDB - Normal subgroup $C_4^2 \trianglelefteq C_4^2.C

Subgroup ($H$) information

Description:	$C_4^2$
Order:	$16$$\medspace = 2^{4} $
Index:	$16$$\medspace = 2^{4} $
Exponent:	$4$$\medspace = 2^{2} $
Generators:	$be, c^{2}$ be, c^{2}
Nilpotency class:	$1$
Derived length:	$1$

The subgroup is characteristic (hence 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:	$C_4^2.C_2^4$
Order:	$256$$\medspace = 2^{8} $
Exponent:	$8$$\medspace = 2^{3} $
Nilpotency class:	$3$
Derived length:	$2$

The ambient group is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metabelian.

Quotient group ($Q$) structure

Description:	$C_2^4$
Order:	$16$$\medspace = 2^{4} $
Exponent:	$2$
Automorphism Group:	$A_8$, of order $20160$$\medspace = 2^{6} \cdot 3^{2} \cdot 5 \cdot 7 $
Outer Automorphisms:	$A_8$, of order $20160$$\medspace = 2^{6} \cdot 3^{2} \cdot 5 \cdot 7 $
Nilpotency class:	$1$
Derived length:	$1$

The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), and rational.

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 $W = G / Z_G(H)$.

$\operatorname{Aut}(G)$	$C_2^9\times D_4$
$\operatorname{Aut}(H)$	$\GL(2,\mathbb{Z}/4)$, of order $96$$\medspace = 2^{5} \cdot 3 $
$\operatorname{res}(\operatorname{Aut}(G))$	$C_2^3$, of order $8$$\medspace = 2^{3} $
$\card{\operatorname{ker}(\operatorname{res})}$	$512$$\medspace = 2^{9} $
$W$	$C_2^2$, of order $4$$\medspace = 2^{2} $