This group is not stored in the database. However, basic information about the group, computed on the fly, is listed below.
Group information
| Description: | $C_{2} \times C_{4004}$ |
| Order: | \(8008\)\(\medspace = 2^{3} \cdot 7 \cdot 11 \cdot 13 \) |
| Exponent: | \(4004\)\(\medspace = 2^{2} \cdot 7 \cdot 11 \cdot 13 \) |
| Automorphism group: | Group of order \(5760\)\(\medspace = 2^{7} \cdot 3^{2} \cdot 5 \) |
| Outer automorphisms: | Group of order \(5760\)\(\medspace = 2^{7} \cdot 3^{2} \cdot 5 \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
This group is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group). Whether it is metacyclic or rational has not been computed.
Group statistics
| Order | 1 | 2 | 4 | 7 | 11 | 13 | 14 | 22 | 26 | 28 | 44 | 52 | 77 | 91 | 143 | 154 | 182 | 286 | 308 | 364 | 572 | 1001 | 2002 | 4004 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Elements | 1 | 3 | 4 | 6 | 10 | 12 | 18 | 30 | 36 | 24 | 40 | 48 | 60 | 72 | 120 | 180 | 216 | 360 | 240 | 288 | 480 | 720 | 2160 | 2880 | 8008 |
| Conjugacy classes | 1 | 3 | 4 | 6 | 10 | 12 | 18 | 30 | 36 | 24 | 40 | 48 | 60 | 72 | 120 | 180 | 216 | 360 | 240 | 288 | 480 | 720 | 2160 | 2880 | 8008 |
| Divisions | data not computed | ||||||||||||||||||||||||
| Autjugacy classes | data not computed | ||||||||||||||||||||||||
| Dimension | 1 | |
|---|---|---|
| Irr. complex chars. | 8008 | 8008 |
Constructions
| Rank: | $2$ |
| Inequivalent generating pairs: | not computed |
Homology
| Primary decomposition: | $C_{2} \times C_{4} \times C_{7} \times C_{11} \times C_{13}$ |
Subgroups
| Center: | $Z \simeq$ $C_{2} \times C_{4004}$ | $G/Z \simeq$ $C_1$ | |
| Commutator: | $G' \simeq$ $C_1$ | $G/G' \simeq$ $C_{2} \times C_{4004}$ | |
| Frattini: | $\Phi \simeq$ $C_2$ | $G/\Phi \simeq$ $C_{4004}$ | |
| Fitting: | $\operatorname{Fit} \simeq$ $C_{2} \times C_{4004}$ | $G/\operatorname{Fit} \simeq$ $C_1$ | |
| Radical: | $R \simeq$ $C_{2} \times C_{4004}$ | $G/R \simeq$ $C_1$ | |
| Socle: | $S \simeq$ $C_{4004}$ | $G/S \simeq$ $C_2$ | |
| 2-Sylow subgroup: | $P_{2} \simeq$ $C_2\times C_4$ | ||
| 7-Sylow subgroup: | $P_{7} \simeq$ $C_7$ | ||
| 11-Sylow subgroup: | $P_{11} \simeq$ $C_{11}$ | ||
| 13-Sylow subgroup: | $P_{13} \simeq$ $C_{13}$ |