Subgroup ($H$) information
| Description: | $C_{29}:C_{14}$ |
| Order: | \(406\)\(\medspace = 2 \cdot 7 \cdot 29 \) |
| Index: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(406\)\(\medspace = 2 \cdot 7 \cdot 29 \) |
| Generators: |
$a^{14}, a^{4}, b^{2}$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), nonabelian, and a Z-group (hence solvable, supersolvable, monomial, metacyclic, 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: | $C_2^3$ |
| Order: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(2\) |
| Automorphism Group: | $\PSL(2,7)$, of order \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| Outer Automorphisms: | $\PSL(2,7)$, of order \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $S_4\times F_{29}$, of order \(19488\)\(\medspace = 2^{5} \cdot 3 \cdot 7 \cdot 29 \) |
| $\operatorname{Aut}(H)$ | $F_{29}$, of order \(812\)\(\medspace = 2^{2} \cdot 7 \cdot 29 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $F_{29}$, of order \(812\)\(\medspace = 2^{2} \cdot 7 \cdot 29 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| $W$ | $F_{29}$, of order \(812\)\(\medspace = 2^{2} \cdot 7 \cdot 29 \) |
Related subgroups
Other information
| Möbius function | $-8$ |
| Projective image | $C_2^2\times F_{29}$ |