Subgroup ($H$) information
| Description: | $C_{43}$ |
| Order: | \(43\) |
| Index: | \(40\)\(\medspace = 2^{3} \cdot 5 \) |
| Exponent: | \(43\) |
| Generators: |
$b^{20}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), a direct factor, cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), central, a $43$-Sylow subgroup (hence a Hall subgroup), a $p$-group, and simple.
Ambient group ($G$) information
| Description: | $D_4\times C_{215}$ |
| Order: | \(1720\)\(\medspace = 2^{3} \cdot 5 \cdot 43 \) |
| Exponent: | \(860\)\(\medspace = 2^{2} \cdot 5 \cdot 43 \) |
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The ambient group is nonabelian, elementary for $p = 2$ (hence nilpotent, solvable, supersolvable, monomial, and hyperelementary), and metacyclic (hence metabelian).
Quotient group ($Q$) structure
| Description: | $C_5\times D_4$ |
| Order: | \(40\)\(\medspace = 2^{3} \cdot 5 \) |
| Exponent: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Automorphism Group: | $C_4\times D_4$, of order \(32\)\(\medspace = 2^{5} \) |
| Outer Automorphisms: | $C_2\times C_4$, of order \(8\)\(\medspace = 2^{3} \) |
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The quotient is nonabelian, elementary for $p = 2$ (hence nilpotent, solvable, supersolvable, monomial, and hyperelementary), and metacyclic (hence metabelian).
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2\times D_4\times C_{84}$, of order \(1344\)\(\medspace = 2^{6} \cdot 3 \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $C_{42}$, of order \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_{42}$, of order \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(32\)\(\medspace = 2^{5} \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $D_4\times C_{215}$ | |||
| Normalizer: | $D_4\times C_{215}$ | |||
| Complements: | $C_5\times D_4$ | |||
| Minimal over-subgroups: | $C_{215}$ | $C_{86}$ | $C_{86}$ | $C_{86}$ |
| Maximal under-subgroups: | $C_1$ |
Other information
| Möbius function | $0$ |
| Projective image | $C_5\times D_4$ |