This group is not stored in the database. However, basic information about the group, computed on the fly, is listed below.
Group information
| Description: | $C_{2} \times C_{3420}$ | |
| Order: | \(6840\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \cdot 19 \) | |
| Exponent: | \(3420\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5 \cdot 19 \) | |
| Automorphism group: | Group of order 3456 | |
| 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 | 5 | 6 | 9 | 10 | 12 | 15 | 18 | 19 | 20 | 30 | 36 | 38 | 45 | 57 | 60 | 76 | 90 | 95 | 114 | 171 | 180 | 190 | 228 | 285 | 342 | 380 | 570 | 684 | 855 | 1140 | 1710 | 3420 | ||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Elements | 1 | 3 | 2 | 4 | 4 | 6 | 6 | 12 | 8 | 8 | 18 | 18 | 16 | 24 | 24 | 54 | 24 | 36 | 32 | 72 | 72 | 72 | 108 | 108 | 96 | 216 | 144 | 144 | 324 | 288 | 432 | 432 | 432 | 576 | 1296 | 1728 | 6840 | |
| Conjugacy classes | 1 | 3 | 2 | 4 | 4 | 6 | 6 | 12 | 8 | 8 | 18 | 18 | 16 | 24 | 24 | 54 | 24 | 36 | 32 | 72 | 72 | 72 | 108 | 108 | 96 | 216 | 144 | 144 | 324 | 288 | 432 | 432 | 432 | 576 | 1296 | 1728 | 6840 | |
| Divisions | data not computed | |||||||||||||||||||||||||||||||||||||
| Autjugacy classes | data not computed | |||||||||||||||||||||||||||||||||||||
| Dimension | 1 | |
|---|---|---|
| Irr. complex chars. | 6840 | 6840 |
Constructions
| Rank: | $2$ |
| Inequivalent generating pairs: | not computed |
Homology
| Primary decomposition: | $C_{2} \times C_{4} \times C_{9} \times C_{5} \times C_{19}$ |
Subgroups
| Center: | $Z \simeq$ $C_{2} \times C_{3420}$ | $G/Z \simeq$ $C_1$ | |
| Commutator: | $G' \simeq$ $C_1$ | $G/G' \simeq$ $C_{2} \times C_{3420}$ | |
| Frattini: | $\Phi \simeq$ $C_6$ | $G/\Phi \simeq$ $C_2\times C_{570}$ | |
| Fitting: | $\operatorname{Fit} \simeq$ $C_{2} \times C_{3420}$ | $G/\operatorname{Fit} \simeq$ $C_1$ | |
| Radical: | $R \simeq$ $C_{2} \times C_{3420}$ | $G/R \simeq$ $C_1$ | |
| Socle: | $S \simeq$ $C_2\times C_{570}$ | $G/S \simeq$ $C_6$ | |
| 2-Sylow subgroup: | $P_{2} \simeq$ $C_2\times C_4$ | ||
| 3-Sylow subgroup: | $P_{3} \simeq$ $C_9$ | ||
| 5-Sylow subgroup: | $P_{5} \simeq$ $C_5$ | ||
| 19-Sylow subgroup: | $P_{19} \simeq$ $C_{19}$ |