Subgroup ($H$) information
| Description: | $C_8$ |
| Order: | \(8\)\(\medspace = 2^{3} \) |
| Index: | \(232\)\(\medspace = 2^{3} \cdot 29 \) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Generators: |
$b^{4}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), and a $p$-group.
Ambient group ($G$) information
| Description: | $C_{29}:Q_{64}$ |
| Order: | \(1856\)\(\medspace = 2^{6} \cdot 29 \) |
| Exponent: | \(928\)\(\medspace = 2^{5} \cdot 29 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
Quotient group ($Q$) structure
| Description: | $C_{29}:D_4$ |
| Order: | \(232\)\(\medspace = 2^{3} \cdot 29 \) |
| Exponent: | \(116\)\(\medspace = 2^{2} \cdot 29 \) |
| Automorphism Group: | $C_2^2\times F_{29}$, of order \(3248\)\(\medspace = 2^{4} \cdot 7 \cdot 29 \) |
| Outer Automorphisms: | $C_{28}$, of order \(28\)\(\medspace = 2^{2} \cdot 7 \) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{232}.C_{14}.C_2^3.C_2^3$ |
| $\operatorname{Aut}(H)$ | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(51968\)\(\medspace = 2^{8} \cdot 7 \cdot 29 \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
| Centralizer: | $C_{29}:C_{32}$ | |||
| Normalizer: | $C_{29}:Q_{64}$ | |||
| Minimal over-subgroups: | $C_{232}$ | $C_{16}$ | $Q_{16}$ | $Q_{16}$ |
| Maximal under-subgroups: | $C_4$ |
Other information
| Möbius function | $0$ |
| Projective image | $C_{29}:D_{16}$ |