Subgroup ($H$) information
| Description: | $C_7\times C_{35}$ |
| Order: | \(245\)\(\medspace = 5 \cdot 7^{2} \) |
| Index: | \(2\) |
| Exponent: | \(35\)\(\medspace = 5 \cdot 7 \) |
| Generators: |
$c^{21}, c^{5}, b$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the commutator subgroup (hence characteristic and normal), the Fitting subgroup (hence nilpotent, solvable, supersolvable, and monomial), the socle, maximal, a semidirect factor, abelian (hence metabelian and an A-group), a Hall subgroup, elementary for $p = 7$ (hence hyperelementary), and metacyclic.
Ambient group ($G$) information
| Description: | $C_7:D_{35}$ |
| Order: | \(490\)\(\medspace = 2 \cdot 5 \cdot 7^{2} \) |
| Exponent: | \(70\)\(\medspace = 2 \cdot 5 \cdot 7 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
Quotient group ($Q$) structure
| Description: | $C_2$ |
| Order: | \(2\) |
| Exponent: | \(2\) |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group, simple, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $F_5\times C_7^2:C_3.\SL(2,7).C_2$ |
| $\operatorname{Aut}(H)$ | $C_4\times \GL(2,7)$, of order \(8064\)\(\medspace = 2^{7} \cdot 3^{2} \cdot 7 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_4\times \GL(2,7)$, of order \(8064\)\(\medspace = 2^{7} \cdot 3^{2} \cdot 7 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(245\)\(\medspace = 5 \cdot 7^{2} \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
Other information
| Möbius function | $-1$ |
| Projective image | $C_7:D_{35}$ |