Non-normal subgroup $D_{16}:D_4 \subseteq C_3\times \SL(2,7).D_8$

• Label: 16128.bb.63.a1.a1
• Order: $ 2^{8} $
• Index: $ 3^{2} \cdot 7 $
• Normal: No

Subgroup ($H$) information

Description:	$D_{16}:D_4$
Order:	\(256\)\(\medspace = 2^{8} \)
Index:	\(63\)\(\medspace = 3^{2} \cdot 7 \)
Exponent:	\(16\)\(\medspace = 2^{4} \)
Generators:	$\left(\begin{array}{rrrr} 5 & 2 & 3 & 5 \\ 4 & 1 & 4 & 3 \\ 6 & 0 & 6 & 5 \\ 5 & 6 & 3 & 2 \end{array}\right), \left(\begin{array}{rrrr} 6 & 2 & 4 & 0 \\ 5 & 1 & 0 & 3 \\ 1 & 0 & 1 & 2 \\ 0 & 6 & 5 & 6 \end{array}\right), \left(\begin{array}{rrrr} 1 & 1 & 5 & 6 \\ 2 & 5 & 4 & 5 \\ 6 & 2 & 5 & 6 \\ 0 & 5 & 0 & 3 \end{array}\right), \left(\begin{array}{rrrr} 1 & 4 & 5 & 0 \\ 3 & 1 & 3 & 0 \\ 4 & 3 & 0 & 1 \\ 0 & 2 & 5 & 5 \end{array}\right), \left(\begin{array}{rrrr} 6 & 0 & 0 & 0 \\ 0 & 6 & 0 & 0 \\ 0 & 0 & 6 & 0 \\ 0 & 0 & 0 & 6 \end{array}\right), \left(\begin{array}{rrrr} 5 & 4 & 6 & 3 \\ 1 & 4 & 1 & 6 \\ 5 & 0 & 0 & 3 \\ 3 & 5 & 6 & 6 \end{array}\right), \left(\begin{array}{rrrr} 2 & 1 & 4 & 5 \\ 4 & 4 & 6 & 1 \\ 2 & 3 & 5 & 5 \\ 5 & 4 & 6 & 3 \end{array}\right), \left(\begin{array}{rrrr} 6 & 4 & 6 & 3 \\ 1 & 5 & 1 & 6 \\ 5 & 0 & 1 & 3 \\ 3 & 5 & 6 & 0 \end{array}\right)$
Nilpotency class:	$4$
Derived length:	$2$

The subgroup is nonabelian, a $2$-Sylow subgroup (hence nilpotent, solvable, supersolvable, a Hall subgroup, and monomial), a $p$-group (hence elementary and hyperelementary), and metabelian.

Ambient group ($G$) information

Description:	$C_3\times \SL(2,7).D_8$
Order:	\(16128\)\(\medspace = 2^{8} \cdot 3^{2} \cdot 7 \)
Exponent:	\(336\)\(\medspace = 2^{4} \cdot 3 \cdot 7 \)
Derived length:	$2$

The ambient group is nonabelian and nonsolvable.

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 $W = N_G(H) / Z_G(H)$.

$\operatorname{Aut}(G)$	$(C_2\times C_4).C_2^4.\SO(3,7)$
$\operatorname{Aut}(H)$	$C_4^2.C_2^3.C_2^4$
$W$	$D_4\times D_8$, of order \(128\)\(\medspace = 2^{7} \)

Related subgroups

Centralizer:

$C_6$

Normalizer:

$C_{12}.D_4^2$

Normal closure:

$\SL(2,7).D_8$

Core:

$\SD_{32}$

Minimal over-subgroups:

$C_{12}.D_4^2$

$\GL(2,3).D_8$

$\GL(2,3).D_8$

Maximal under-subgroups:

$D_{16}:C_4$

$D_8:D_4$

$D_4.D_8$

$D_4.D_8$

$D_{16}:C_2^2$

$D_{16}:C_2^2$

$C_2.D_4^2$

$C_8.D_8$

$C_{16}:D_4$

$C_{16}.D_4$

$C_{16}.D_4$

$D_4.D_8$

$D_4.D_8$

$D_8:D_4$

$C_{16}.D_4$

Other information

Number of subgroups in this conjugacy class	$21$
Möbius function	not computed
Projective image	not computed