LMFDB - Normal subgroup $C_5 \trianglelefteq C_2^3:F_5\times S

Subgroup ($H$) information

Description:	$C_5$
Order:	$5$
Index:	$768$$\medspace = 2^{8} \cdot 3 $
Exponent:	$5$
Generators:	$\langle(1,5,3,2,4)\rangle$ (1,5,3,2,4)
Nilpotency class:	$1$
Derived length:	$1$

The subgroup is characteristic (hence normal), a semidirect factor, cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $5$-Sylow subgroup (hence a Hall subgroup), a $p$-group, and simple.

Ambient group ($G$) information

Description:	$C_2^3:F_5\times S_4$
Order:	$3840$$\medspace = 2^{8} \cdot 3 \cdot 5 $
Exponent:	$60$$\medspace = 2^{2} \cdot 3 \cdot 5 $
Derived length:	$3$

The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.

Quotient group ($Q$) structure

Description:	$C_2^6.D_6$
Order:	$768$$\medspace = 2^{8} \cdot 3 $
Exponent:	$12$$\medspace = 2^{2} \cdot 3 $
Automorphism Group:	$(C_2^4\times A_4).C_2^6.C_2^3$
Outer Automorphisms:	$C_2^6.C_2^4$, of order $1024$$\medspace = 2^{10} $
Nilpotency class:	$-1$
Derived length:	$3$

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 $W = G / Z_G(H)$.

$\operatorname{Aut}(G)$	$(C_5\times A_4).C_2^6.C_2^5$
$\operatorname{Aut}(H)$	$C_4$, of order $4$$\medspace = 2^{2} $
$\operatorname{res}(\operatorname{Aut}(G))$	$C_4$, of order $4$$\medspace = 2^{2} $
$\card{\operatorname{ker}(\operatorname{res})}$	$30720$$\medspace = 2^{11} \cdot 3 \cdot 5 $
$W$	$C_4$, of order $4$$\medspace = 2^{2} $