Subgroup ($H$) information
| Description: | $C_{21}$ |
| Order: | \(21\)\(\medspace = 3 \cdot 7 \) |
| Index: | \(507\)\(\medspace = 3 \cdot 13^{2} \) |
| Exponent: | \(21\)\(\medspace = 3 \cdot 7 \) |
| Generators: |
$c^{13}, a^{3}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the center (hence characteristic, normal, abelian, central, nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a direct factor, and cyclic (hence elementary ($p = 3,7$), hyperelementary, metacyclic, and a Z-group).
Ambient group ($G$) information
| Description: | $C_3\times C_{13}^2:C_{21}$ |
| Order: | \(10647\)\(\medspace = 3^{2} \cdot 7 \cdot 13^{2} \) |
| Exponent: | \(273\)\(\medspace = 3 \cdot 7 \cdot 13 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
Quotient group ($Q$) structure
| Description: | $C_{13}^2:C_3$ |
| Order: | \(507\)\(\medspace = 3 \cdot 13^{2} \) |
| Exponent: | \(39\)\(\medspace = 3 \cdot 13 \) |
| Automorphism Group: | $C_{13}^2.\GL(2,13)$, of order \(4429152\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 7 \cdot 13^{3} \) |
| Outer Automorphisms: | $\SL(2,13):C_4$, of order \(8736\)\(\medspace = 2^{5} \cdot 3 \cdot 7 \cdot 13 \) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, supersolvable (hence solvable and monomial), metabelian, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_6\times S_3\times C_{13}^2.C_{12}.\PSL(2,13).C_2$ |
| $\operatorname{Aut}(H)$ | $C_2\times C_6$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_3\times C_{13}^2:C_{21}$ | |
| Normalizer: | $C_3\times C_{13}^2:C_{21}$ | |
| Complements: | $C_{13}^2:C_3$ | |
| Minimal over-subgroups: | $C_{273}$ | $C_3\times C_{21}$ |
| Maximal under-subgroups: | $C_7$ | $C_3$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $-2197$ |
| Projective image | $C_{13}^2:C_3$ |