Subgroup ($H$) information
| Description: | $C_{14}$ |
| Order: | \(14\)\(\medspace = 2 \cdot 7 \) |
| Index: | \(154\)\(\medspace = 2 \cdot 7 \cdot 11 \) |
| Exponent: | \(14\)\(\medspace = 2 \cdot 7 \) |
| Generators: |
$a^{14}, b^{22}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal) and cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,7$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
Ambient group ($G$) information
| Description: | $C_{77}:C_{28}$ |
| Order: | \(2156\)\(\medspace = 2^{2} \cdot 7^{2} \cdot 11 \) |
| Exponent: | \(308\)\(\medspace = 2^{2} \cdot 7 \cdot 11 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and an A-group.
Quotient group ($Q$) structure
| Description: | $C_7\times D_{11}$ |
| Order: | \(154\)\(\medspace = 2 \cdot 7 \cdot 11 \) |
| Exponent: | \(154\)\(\medspace = 2 \cdot 7 \cdot 11 \) |
| Automorphism Group: | $C_6\times F_{11}$, of order \(660\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 11 \) |
| Outer Automorphisms: | $C_{30}$, of order \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{77}.C_{15}.C_6.C_2^3$ |
| $\operatorname{Aut}(H)$ | $C_6$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_6$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(9240\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
| Centralizer: | $C_7\times C_{154}$ | ||
| Normalizer: | $C_{77}:C_{28}$ | ||
| Minimal over-subgroups: | $C_{154}$ | $C_7\times C_{14}$ | $C_7:C_4$ |
| Maximal under-subgroups: | $C_7$ | $C_2$ |
Other information
| Möbius function | $-11$ |
| Projective image | $C_7\times D_{77}$ |