Subgroup ($H$) information
| Description: | $C_7^2$ |
| Order: | \(49\)\(\medspace = 7^{2} \) |
| Index: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Exponent: | \(7\) |
| Generators: |
$b, c^{5}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), a semidirect factor, abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $7$-Sylow subgroup (hence a Hall subgroup), a $p$-group (hence elementary and 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: | $D_5$ |
| Order: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Exponent: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Automorphism Group: | $F_5$, of order \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $F_5\times C_7^2:C_3.\SL(2,7).C_2$ |
| $\operatorname{Aut}(H)$ | $\GL(2,7)$, of order \(2016\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 7 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $\GL(2,7)$, of order \(2016\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 7 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(980\)\(\medspace = 2^{2} \cdot 5 \cdot 7^{2} \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
| Centralizer: | $C_7\times C_{35}$ | |||||||
| Normalizer: | $C_7:D_{35}$ | |||||||
| Complements: | $D_5$ | |||||||
| Minimal over-subgroups: | $C_7\times C_{35}$ | $C_7:D_7$ | ||||||
| Maximal under-subgroups: | $C_7$ | $C_7$ | $C_7$ | $C_7$ | $C_7$ | $C_7$ | $C_7$ | $C_7$ |
Other information
| Möbius function | $5$ |
| Projective image | $C_7:D_{35}$ |