LMFDB - Normal subgroup $Q_8\times D_{14} \trianglelefteq \SD_{16}:D

Subgroup ($H$) information

Description:	$Q_8\times D_{14}$
Order:	$224$$\medspace = 2^{5} \cdot 7 $
Index:	$2$
Exponent:	$28$$\medspace = 2^{2} \cdot 7 $
Generators:	$ad^{21}, d^{8}, c, d^{42}, b, d^{28}$ ad^{21}, d^{8}, c, d^{42}, b, d^{28}
Derived length:	$2$

The subgroup is characteristic (hence normal), maximal, a semidirect factor, nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.

Ambient group ($G$) information

Description:	$\SD_{16}:D_{14}$
Order:	$448$$\medspace = 2^{6} \cdot 7 $
Exponent:	$56$$\medspace = 2^{3} \cdot 7 $
Derived length:	$2$

The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.

Quotient group ($Q$) structure

Description:	$C_2$
Order:	$2$
Exponent:	$2$
Automorphism Group:	$C_1$, of order $1$
Outer Automorphisms:	$C_1$, of order $1$
Derived length:	$1$

The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group, simple, 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 $W = G / Z_G(H)$.

$\operatorname{Aut}(G)$	$C_7.(C_2^4\times C_6).C_2^5$
$\operatorname{Aut}(H)$	$C_2\wr D_6.F_7$, of order $32256$$\medspace = 2^{9} \cdot 3^{2} \cdot 7 $
$\card{W}$	$112$$\medspace = 2^{4} \cdot 7 $