LMFDB - Normal subgroup $C_{60} \trianglelefteq C_{60}.C

Subgroup ($H$) information

Description:	$C_{60}$
Order:	$60$$\medspace = 2^{2} \cdot 3 \cdot 5 $
Index:	$16$$\medspace = 2^{4} $
Exponent:	$60$$\medspace = 2^{2} \cdot 3 \cdot 5 $
Generators:	$c^{6}, d^{2}, c^{12}, c^{8}$ c^{6}, d^{2}, c^{12}, c^{8}
Nilpotency class:	$1$
Derived length:	$1$

The subgroup is the commutator subgroup (hence characteristic and normal) and cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,3,5$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).

Ambient group ($G$) information

Description:	$C_{60}.C_2^4$
Order:	$960$$\medspace = 2^{6} \cdot 3 \cdot 5 $
Exponent:	$120$$\medspace = 2^{3} \cdot 3 \cdot 5 $
Derived length:	$2$

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

Quotient group ($Q$) structure

Description:	$C_2^4$
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 $W = G / Z_G(H)$.

$\operatorname{Aut}(G)$	$C_{15}:(C_2^3.C_2^6.C_2)$
$\operatorname{Aut}(H)$	$C_2^2\times C_4$, of order $16$$\medspace = 2^{4} $
$\operatorname{res}(\operatorname{Aut}(G))$	$C_2^2\times C_4$, of order $16$$\medspace = 2^{4} $
$\card{\operatorname{ker}(\operatorname{res})}$	$960$$\medspace = 2^{6} \cdot 3 \cdot 5 $
$W$	$C_2^3$, of order $8$$\medspace = 2^{3} $