Subgroup ($H$) information
| Description: | $C_{215}$ |
| Order: | \(215\)\(\medspace = 5 \cdot 43 \) |
| Index: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Exponent: | \(215\)\(\medspace = 5 \cdot 43 \) |
| Generators: |
$a^{2}b^{258}, b^{10}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is normal, a direct factor, cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 5,43$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), and central.
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: | $C_2\times C_{10}$ |
| Order: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Exponent: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Automorphism Group: | $C_4\times S_3$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Outer Automorphisms: | $C_4\times S_3$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), elementary for $p = 2$ (hence hyperelementary), and metacyclic.
Automorphism information
While the subgroup $H$ is not characteristic, the stabilizer $S$ of $H$ in the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : S \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphisms $\operatorname{Inn}(G) \cap S$ is the Weyl group $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{42}\times S_3\times \GL(2,5)$ |
| $\operatorname{Aut}(H)$ | $C_2\times C_{84}$, of order \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| $\operatorname{res}(S)$ | $C_2\times C_{84}$, of order \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
Other information
| Möbius function | $-2$ |
| Projective image | $C_2\times C_{10}$ |