Subgroup ($H$) information
| Description: | $C_{58}:C_{14}$ |
| Order: | \(812\)\(\medspace = 2^{2} \cdot 7 \cdot 29 \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(406\)\(\medspace = 2 \cdot 7 \cdot 29 \) |
| Generators: |
$b^{29}, c, b^{2}, a^{4}$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), a semidirect factor, nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), 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: | $C_4$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Automorphism Group: | $C_2$, of order \(2\) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group) and a $p$-group.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $S_4\times F_{29}$, of order \(19488\)\(\medspace = 2^{5} \cdot 3 \cdot 7 \cdot 29 \) |
| $\operatorname{Aut}(H)$ | $S_3\times F_{29}$, of order \(4872\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \cdot 29 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $S_3\times F_{29}$, of order \(4872\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \cdot 29 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(4\)\(\medspace = 2^{2} \) |
| $W$ | $F_{29}$, of order \(812\)\(\medspace = 2^{2} \cdot 7 \cdot 29 \) |
Related subgroups
Other information
| Möbius function | $0$ |
| Projective image | $F_{29}$ |