Subgroup ($H$) information
| Description: | $C_2$ |
| Order: | \(2\) |
| Index: | \(768\)\(\medspace = 2^{8} \cdot 3 \) |
| Exponent: | \(2\) |
| Generators: |
$\left(\begin{array}{rr}
15 & 8 \\
8 & 7
\end{array}\right)$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is normal, a direct factor, cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), central, a $p$-group, simple, and rational.
Ambient group ($G$) information
| Description: | $C_4^4:C_6$ |
| Order: | \(1536\)\(\medspace = 2^{9} \cdot 3 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, monomial (hence solvable), metabelian, and an A-group.
Quotient group ($Q$) structure
| Description: | $C_4^3:C_{12}$ |
| Order: | \(768\)\(\medspace = 2^{8} \cdot 3 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Automorphism Group: | $C_4^2:C_3.D_4\times \GL(2,\mathbb{Z}/4)$ |
| Outer Automorphisms: | $D_4\times \GL(2,\mathbb{Z}/4)$, of order \(768\)\(\medspace = 2^{8} \cdot 3 \) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, monomial (hence solvable), metabelian, and an A-group.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_4^2:C_3.D_4\times C_2^6.S_4$ |
| $\operatorname{Aut}(H)$ | $C_1$, of order $1$ |
| $\operatorname{res}(S)$ | $C_1$, of order $1$ |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(147456\)\(\medspace = 2^{14} \cdot 3^{2} \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_4^4:C_6$ | |||
| Normalizer: | $C_4^4:C_6$ | |||
| Complements: | $C_4^3:C_{12}$ | |||
| Minimal over-subgroups: | $C_6$ | $C_2^2$ | $C_2^2$ | $C_2^2$ |
| Maximal under-subgroups: | $C_1$ |
Other information
| Number of subgroups in this autjugacy class | $4$ |
| Number of conjugacy classes in this autjugacy class | $4$ |
| Möbius function | $0$ |
| Projective image | $C_4^3:C_{12}$ |