Subgroup ($H$) information
| Description: | $C_5\times C_{215}$ |
| Order: | \(1075\)\(\medspace = 5^{2} \cdot 43 \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(215\)\(\medspace = 5 \cdot 43 \) |
| Generators: |
$a^{2}, b^{172}, b^{10}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), a direct factor, central (hence abelian, nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a Hall subgroup, elementary for $p = 5$ (hence hyperelementary), and metacyclic.
Ambient group ($G$) information
| Description: | $C_{10}\times C_{430}$ |
| Order: | \(4300\)\(\medspace = 2^{2} \cdot 5^{2} \cdot 43 \) |
| Exponent: | \(430\)\(\medspace = 2 \cdot 5 \cdot 43 \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The ambient group is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group) and metacyclic.
Quotient group ($Q$) structure
| Description: | $C_2^2$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(2\) |
| Automorphism Group: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Outer Automorphisms: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| 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), metacyclic, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{42}\times S_3\times \GL(2,5)$ |
| $\operatorname{Aut}(H)$ | $C_{42}\times \GL(2,5)$ |
| $\card{\operatorname{res}(\operatorname{Aut}(G))}$ | \(20160\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 5 \cdot 7 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(6\)\(\medspace = 2 \cdot 3 \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
Other information
| Möbius function | $2$ |
| Projective image | $C_2^2$ |