LMFDB - Normal subgroup $C_{10} \trianglelefteq D_{12}:D

Subgroup ($H$) information

Description:	$C_{10}$
Order:	$10$$\medspace = 2 \cdot 5 $
Index:	$48$$\medspace = 2^{4} \cdot 3 $
Exponent:	$10$$\medspace = 2 \cdot 5 $
Generators:	$d^{30}, d^{12}$ d^{30}, d^{12}
Nilpotency class:	$1$
Derived length:	$1$

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

Ambient group ($G$) information

Description:	$D_{12}:D_{10}$
Order:	$480$$\medspace = 2^{5} \cdot 3 \cdot 5 $
Exponent:	$60$$\medspace = 2^{2} \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:	$C_2^2\times D_6$
Order:	$48$$\medspace = 2^{4} \cdot 3 $
Exponent:	$6$$\medspace = 2 \cdot 3 $
Automorphism Group:	$S_3\times C_2^3:\GL(3,2)$, of order $8064$$\medspace = 2^{7} \cdot 3^{2} \cdot 7 $
Outer Automorphisms:	$C_2^3:\GL(3,2)$, of order $1344$$\medspace = 2^{6} \cdot 3 \cdot 7 $
Nilpotency class:	$-1$
Derived length:	$2$

The quotient is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, metabelian, an A-group, 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 $W = G / Z_G(H)$.

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