LMFDB - Non-normal subgroup $C_{12}.C_2^4 \subseteq C_{84}.C

Subgroup ($H$) information

Description:	$C_{12}.C_2^4$
Order:	$192$$\medspace = 2^{6} \cdot 3 $
Index:	$7$
Exponent:	$24$$\medspace = 2^{3} \cdot 3 $
Generators:	$a, d^{42}, c^{2}, b, d^{28}, c, d^{21}$ a, d^{42}, c^{2}, b, d^{28}, c, d^{21}
Derived length:	$2$

The subgroup is maximal, nonabelian, a Hall subgroup, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.

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.

Automorphism information

While the subgroup $H$ is not characteristic, the stabilizer $S$ of $H$ in the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : S \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphisms $\operatorname{Inn}(G) \cap S$ is the Weyl group $W = N_G(H) / Z_G(H)$.

$\operatorname{Aut}(G)$	$C_{42}.(C_2^5\times C_6).C_2^2$
$\operatorname{Aut}(H)$	$C_2^4:D_6^2$, of order $2304$$\medspace = 2^{8} \cdot 3^{2} $
$\operatorname{res}(S)$	$D_6\times D_4^2$, of order $768$$\medspace = 2^{8} \cdot 3 $
$\card{\operatorname{ker}(\operatorname{res})}$	$6$$\medspace = 2 \cdot 3 $
$W$	$C_2^3:D_6$, of order $96$$\medspace = 2^{5} \cdot 3 $