Subgroup ($H$) information
| Description: | $C_{105}:D_4$ |
| Order: | \(840\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 7 \) |
| Index: | $1$ |
| Exponent: | \(420\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 7 \) |
| Generators: |
$\left(\begin{array}{rr}
0 & 1 \\
1 & 0
\end{array}\right), \left(\begin{array}{rr}
60 & 0 \\
0 & 102
\end{array}\right), \left(\begin{array}{rr}
4 & 0 \\
0 & 49
\end{array}\right), \left(\begin{array}{rr}
144 & 0 \\
0 & 148
\end{array}\right), \left(\begin{array}{rr}
1 & 0 \\
0 & 210
\end{array}\right), \left(\begin{array}{rr}
13 & 0 \\
0 & 65
\end{array}\right)$
|
| Derived length: | $2$ |
The subgroup is the radical (hence characteristic, normal, and solvable), a direct factor, nonabelian, a Hall subgroup, supersolvable (hence monomial), hyperelementary for $p = 2$, and metabelian.
Ambient group ($G$) information
| Description: | $C_{105}:D_4$ |
| Order: | \(840\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 7 \) |
| Exponent: | \(420\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 7 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
Quotient group ($Q$) structure
| Description: | $C_1$ |
| Order: | $1$ |
| Exponent: | $1$ |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $0$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary (for every $p$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group (for every $p$), perfect, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^2\times C_7:(C_2\times C_6\times F_5)$ |
| $\operatorname{Aut}(H)$ | $C_2^2\times C_7:(C_2\times C_6\times F_5)$ |
| $W$ | $D_{70}$, of order \(140\)\(\medspace = 2^{2} \cdot 5 \cdot 7 \) |
Related subgroups
| Centralizer: | $C_6$ | |||||
| Normalizer: | $C_{105}:D_4$ | |||||
| Complements: | $C_1$ | |||||
| Maximal under-subgroups: | $C_2\times C_{210}$ | $C_3\times D_{70}$ | $C_{105}:C_4$ | $C_{35}:D_4$ | $C_{21}:D_4$ | $C_{15}:D_4$ |
Other information
| Möbius function | $1$ |
| Projective image | $D_{70}$ |