Subgroup ($H$) information
| Description: | $C_{21}:C_3$ |
| Order: | \(63\)\(\medspace = 3^{2} \cdot 7 \) |
| Index: | \(22\)\(\medspace = 2 \cdot 11 \) |
| Exponent: | \(21\)\(\medspace = 3 \cdot 7 \) |
| Generators: |
$a^{2}, b^{66}, b^{154}$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), a semidirect factor, nonabelian, a Hall subgroup, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 3$, and an A-group.
Ambient group ($G$) information
| Description: | $C_{21}:C_{66}$ |
| Order: | \(1386\)\(\medspace = 2 \cdot 3^{2} \cdot 7 \cdot 11 \) |
| Exponent: | \(462\)\(\medspace = 2 \cdot 3 \cdot 7 \cdot 11 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and an A-group.
Quotient group ($Q$) structure
| Description: | $C_{22}$ |
| Order: | \(22\)\(\medspace = 2 \cdot 11 \) |
| Exponent: | \(22\)\(\medspace = 2 \cdot 11 \) |
| Automorphism Group: | $C_{10}$, of order \(10\)\(\medspace = 2 \cdot 5 \) |
| Outer Automorphisms: | $C_{10}$, of order \(10\)\(\medspace = 2 \cdot 5 \) |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,11$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{10}\times S_3\times F_7$ |
| $\operatorname{Aut}(H)$ | $S_3\times F_7$, of order \(252\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 7 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_2\times F_7$, of order \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| $W$ | $F_7$, of order \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
Related subgroups
| Centralizer: | $C_{33}$ | |||
| Normalizer: | $C_{21}:C_{66}$ | |||
| Complements: | $C_{22}$ | |||
| Minimal over-subgroups: | $C_{21}:C_{33}$ | $C_{21}:C_6$ | ||
| Maximal under-subgroups: | $C_{21}$ | $C_7:C_3$ | $C_7:C_3$ | $C_3^2$ |
Other information
| Möbius function | $1$ |
| Projective image | $C_{21}:C_{66}$ |