Subgroup ($H$) information
| Description: | $C_2^3\times C_{14}$ |
| Order: | \(112\)\(\medspace = 2^{4} \cdot 7 \) |
| Index: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Exponent: | \(14\)\(\medspace = 2 \cdot 7 \) |
| Generators: |
$\left(\begin{array}{rr}
13 & 0 \\
0 & 13
\end{array}\right), \left(\begin{array}{rr}
43 & 0 \\
0 & 43
\end{array}\right), \left(\begin{array}{rr}
43 & 42 \\
0 & 43
\end{array}\right), \left(\begin{array}{rr}
71 & 0 \\
0 & 71
\end{array}\right), \left(\begin{array}{rr}
1 & 60 \\
0 & 1
\end{array}\right)$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence 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: | $D_6$ |
| Order: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Automorphism Group: | $D_6$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 2$, an A-group, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^8.C_2^4.C_7.C_3^3.C_2^3$ |
| $\operatorname{Aut}(H)$ | $C_6\times A_8$, of order \(120960\)\(\medspace = 2^{7} \cdot 3^{3} \cdot 5 \cdot 7 \) |
| $\card{W}$ | \(2\) |
Related subgroups
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | not computed |