This group is not stored in the database. However, basic information about the group, computed on the fly, is listed below.
Group information
| Description: | $C_{2} \times C_{1044}$ |
| Order: | \(2088\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 29 \) |
| Exponent: | \(1044\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 29 \) |
| Automorphism group: | Group of order \(1344\)\(\medspace = 2^{6} \cdot 3 \cdot 7 \) |
| Outer automorphisms: | Group of order \(1344\)\(\medspace = 2^{6} \cdot 3 \cdot 7 \) |
| 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 | 3 | 4 | 6 | 9 | 12 | 18 | 29 | 36 | 58 | 87 | 116 | 174 | 261 | 348 | 522 | 1044 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Elements | 1 | 3 | 2 | 4 | 6 | 6 | 8 | 18 | 28 | 24 | 84 | 56 | 112 | 168 | 168 | 224 | 504 | 672 | 2088 |
| Conjugacy classes | 1 | 3 | 2 | 4 | 6 | 6 | 8 | 18 | 28 | 24 | 84 | 56 | 112 | 168 | 168 | 224 | 504 | 672 | 2088 |
| Divisions | data not computed | ||||||||||||||||||
| Autjugacy classes | data not computed | ||||||||||||||||||
| Dimension | 1 | |
|---|---|---|
| Irr. complex chars. | 2088 | 2088 |
Constructions
| Rank: | $2$ |
| Inequivalent generating pairs: | not computed |
Homology
| Primary decomposition: | $C_{2} \times C_{4} \times C_{9} \times C_{29}$ |
Subgroups
| Center: | $Z \simeq$ $C_{2} \times C_{1044}$ | $G/Z \simeq$ $C_1$ | |
| Commutator: | $G' \simeq$ $C_1$ | $G/G' \simeq$ $C_{2} \times C_{1044}$ | |
| Frattini: | $\Phi \simeq$ $C_6$ | $G/\Phi \simeq$ $C_2\times C_{174}$ | |
| Fitting: | $\operatorname{Fit} \simeq$ $C_{2} \times C_{1044}$ | $G/\operatorname{Fit} \simeq$ $C_1$ | |
| Radical: | $R \simeq$ $C_{2} \times C_{1044}$ | $G/R \simeq$ $C_1$ | |
| Socle: | $S \simeq$ $C_2\times C_{174}$ | $G/S \simeq$ $C_6$ | |
| 2-Sylow subgroup: | $P_{2} \simeq$ $C_2\times C_4$ | ||
| 3-Sylow subgroup: | $P_{3} \simeq$ $C_9$ | ||
| 29-Sylow subgroup: | $P_{29} \simeq$ $C_{29}$ |