LMFDB - Normal subgroup $(C_2\times D_{14}):D_{12} \trianglelefteq (C_2\times D_{14}):D

Subgroup ($H$) information

Description:	$(C_2\times D_{14}):D_{12}$
Order:	$1344$$\medspace = 2^{6} \cdot 3 \cdot 7 $
Index:	$1$
Exponent:	$84$$\medspace = 2^{2} \cdot 3 \cdot 7 $
Generators:	$a, c^{2}, d^{28}, c, d^{42}, d^{12}, b, d^{21}$ a, c^{2}, d^{28}, c, d^{42}, d^{12}, b, d^{21}
Derived length:	$2$

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

Ambient group ($G$) information

Description:	$(C_2\times D_{14}):D_{12}$
Order:	$1344$$\medspace = 2^{6} \cdot 3 \cdot 7 $
Exponent:	$84$$\medspace = 2^{2} \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:	$C_1$
Order:	$1$
Exponent:	$1$
Automorphism Group:	$C_1$, of order $1$
Outer Automorphisms:	$C_1$, of order $1$
Derived length:	$0$

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

$\operatorname{Aut}(G)$	$C_{42}.(C_2^5\times C_6).C_2^4$
$\operatorname{Aut}(H)$	$C_{42}.(C_2^5\times C_6).C_2^4$
$\card{W}$	$336$$\medspace = 2^{4} \cdot 3 \cdot 7 $