LMFDB - Normal subgroup $C_{42} \trianglelefteq C_{84}.C

Subgroup ($H$) information

Description:	$C_{42}$
Order:	$42$$\medspace = 2 \cdot 3 \cdot 7 $
Index:	$32$$\medspace = 2^{5} $
Exponent:	$42$$\medspace = 2 \cdot 3 \cdot 7 $
Generators:	$d^{42}, d^{12}, d^{28}$ d^{42}, d^{12}, d^{28}
Nilpotency class:	$1$
Derived length:	$1$

The subgroup is the socle (hence characteristic and normal) and cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,3,7$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).

Ambient group ($G$) information

Description:	$C_{84}.C_2^4$
Order:	$1344$$\medspace = 2^{6} \cdot 3 \cdot 7 $
Exponent:	$168$$\medspace = 2^{3} \cdot 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:	$C_2^2\times D_4$
Order:	$32$$\medspace = 2^{5} $
Exponent:	$4$$\medspace = 2^{2} $
Automorphism Group:	$C_2^6:(C_2\times S_4)$, of order $3072$$\medspace = 2^{10} \cdot 3 $
Outer Automorphisms:	$C_2^5:S_4$, of order $768$$\medspace = 2^{8} \cdot 3 $
Nilpotency class:	$2$
Derived length:	$2$

The quotient is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), metabelian, 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 $W = G / Z_G(H)$.

$\operatorname{Aut}(G)$	$C_{42}.(C_2^5\times C_6).C_2^2$
$\operatorname{Aut}(H)$	$C_2\times C_6$, of order $12$$\medspace = 2^{2} \cdot 3 $
$\operatorname{res}(\operatorname{Aut}(G))$	$C_2\times C_6$, of order $12$$\medspace = 2^{2} \cdot 3 $
$\card{\operatorname{ker}(\operatorname{res})}$	$2688$$\medspace = 2^{7} \cdot 3 \cdot 7 $
$W$	$C_2^2$, of order $4$$\medspace = 2^{2} $