Subgroup ($H$) information
| Description: | $C_2^2\times C_{42}$ |
| Order: | \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| Index: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
| Generators: |
$\left(\begin{array}{rr}
13 & 0 \\
0 & 13
\end{array}\right), \left(\begin{array}{rr}
43 & 42 \\
0 & 43
\end{array}\right), \left(\begin{array}{rr}
1 & 28 \\
0 & 1
\end{array}\right), \left(\begin{array}{rr}
71 & 0 \\
0 & 71
\end{array}\right), \left(\begin{array}{rr}
1 & 60 \\
0 & 1
\end{array}\right)$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is normal, abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), and elementary for $p = 2$ (hence hyperelementary).
Ambient group ($G$) information
| Description: | $C_2^2\times C_6.D_{28}$ |
| 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: | $C_2\times C_4$ |
| Order: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Automorphism Group: | $D_4$, of order \(8\)\(\medspace = 2^{3} \) |
| Outer Automorphisms: | $D_4$, of order \(8\)\(\medspace = 2^{3} \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), and metacyclic.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^8.C_2^4.C_7.C_3^3.C_2^3$ |
| $\operatorname{Aut}(H)$ | $C_2\times C_6\times \GL(3,2)$, of order \(2016\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 7 \) |
| $\card{W}$ | \(4\)\(\medspace = 2^{2} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $4$ |
| Number of conjugacy classes in this autjugacy class | $4$ |
| Möbius function | not computed |
| Projective image | not computed |