This group is not stored in the database. However, basic information about the group, computed on the fly, is listed below.
Group information
| Description: | $C_{2} \times C_{2002}$ |
| Order: | \(4004\)\(\medspace = 2^{2} \cdot 7 \cdot 11 \cdot 13 \) |
| Exponent: | \(2002\)\(\medspace = 2 \cdot 7 \cdot 11 \cdot 13 \) |
| Automorphism group: | Group of order \(4320\)\(\medspace = 2^{5} \cdot 3^{3} \cdot 5 \) |
| Outer automorphisms: | Group of order \(4320\)\(\medspace = 2^{5} \cdot 3^{3} \cdot 5 \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
This group is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group) and elementary for $p = 2$ (hence hyperelementary). Whether it is metacyclic or rational has not been computed.
Group statistics
| Order | 1 | 2 | 7 | 11 | 13 | 14 | 22 | 26 | 77 | 91 | 143 | 154 | 182 | 286 | 1001 | 2002 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Elements | 1 | 3 | 6 | 10 | 12 | 18 | 30 | 36 | 60 | 72 | 120 | 180 | 216 | 360 | 720 | 2160 | 4004 |
| Conjugacy classes | 1 | 3 | 6 | 10 | 12 | 18 | 30 | 36 | 60 | 72 | 120 | 180 | 216 | 360 | 720 | 2160 | 4004 |
| Divisions | data not computed | ||||||||||||||||
| Autjugacy classes | data not computed | ||||||||||||||||
| Dimension | 1 | |
|---|---|---|
| Irr. complex chars. | 4004 | 4004 |
Constructions
| Rank: | not computed |
| Inequivalent generating tuples: | not computed |
Homology
| Primary decomposition: | $C_{2}^{2} \times C_{7} \times C_{11} \times C_{13}$ |
Subgroups
| Center: | $Z \simeq$ $C_{2} \times C_{2002}$ | $G/Z \simeq$ $C_1$ | |
| Commutator: | $G' \simeq$ $C_1$ | $G/G' \simeq$ $C_{2} \times C_{2002}$ | |
| Frattini: | $\Phi \simeq$ $C_1$ | $G/\Phi \simeq$ $C_{2} \times C_{2002}$ | |
| Fitting: | $\operatorname{Fit} \simeq$ $C_{2} \times C_{2002}$ | $G/\operatorname{Fit} \simeq$ $C_1$ | |
| Radical: | $R \simeq$ $C_{2} \times C_{2002}$ | $G/R \simeq$ $C_1$ | |
| Socle: | $S \simeq$ $C_{2} \times C_{2002}$ | $G/S \simeq$ $C_1$ | |
| 2-Sylow subgroup: | $P_{2} \simeq$ $C_2^2$ | ||
| 7-Sylow subgroup: | $P_{7} \simeq$ $C_7$ | ||
| 11-Sylow subgroup: | $P_{11} \simeq$ $C_{11}$ | ||
| 13-Sylow subgroup: | $P_{13} \simeq$ $C_{13}$ |