Subgroup ($H$) information
| Description: | $C_{58}$ |
| Order: | \(58\)\(\medspace = 2 \cdot 29 \) |
| Index: | \(56\)\(\medspace = 2^{3} \cdot 7 \) |
| Exponent: | \(58\)\(\medspace = 2 \cdot 29 \) |
| Generators: |
$b^{29}c, b^{2}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is normal, a semidirect factor, and cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,29$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
Ambient group ($G$) information
| Description: | $C_2^2\times F_{29}$ |
| Order: | \(3248\)\(\medspace = 2^{4} \cdot 7 \cdot 29 \) |
| Exponent: | \(812\)\(\medspace = 2^{2} \cdot 7 \cdot 29 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
Quotient group ($Q$) structure
| Description: | $C_2\times C_{28}$ |
| Order: | \(56\)\(\medspace = 2^{3} \cdot 7 \) |
| Exponent: | \(28\)\(\medspace = 2^{2} \cdot 7 \) |
| Automorphism Group: | $C_6\times D_4$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Outer Automorphisms: | $C_6\times D_4$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $S_4\times F_{29}$, of order \(19488\)\(\medspace = 2^{5} \cdot 3 \cdot 7 \cdot 29 \) |
| $\operatorname{Aut}(H)$ | $C_{28}$, of order \(28\)\(\medspace = 2^{2} \cdot 7 \) |
| $\operatorname{res}(S)$ | $C_{28}$, of order \(28\)\(\medspace = 2^{2} \cdot 7 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(232\)\(\medspace = 2^{3} \cdot 29 \) |
| $W$ | $C_{28}$, of order \(28\)\(\medspace = 2^{2} \cdot 7 \) |
Related subgroups
Other information
| Möbius function | $0$ |
| Projective image | $C_2\times F_{29}$ |