This group is not stored in the database. However, basic information about the group, computed on the fly, is listed below.
Group information
| Description: | $C_{2} \times C_{6} \times C_{210}$ |
| Order: | \(2520\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \) |
| Exponent: | \(210\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 7 \) |
| Automorphism group: | Group of order \(193536\)\(\medspace = 2^{10} \cdot 3^{3} \cdot 7 \) |
| Outer automorphisms: | Group of order \(193536\)\(\medspace = 2^{10} \cdot 3^{3} \cdot 7 \) |
| 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 | 5 | 6 | 7 | 10 | 14 | 15 | 21 | 30 | 35 | 42 | 70 | 105 | 210 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Elements | 1 | 7 | 8 | 4 | 56 | 6 | 28 | 42 | 32 | 48 | 224 | 24 | 336 | 168 | 192 | 1344 | 2520 |
| Conjugacy classes | 1 | 7 | 8 | 4 | 56 | 6 | 28 | 42 | 32 | 48 | 224 | 24 | 336 | 168 | 192 | 1344 | 2520 |
| Divisions | data not computed | ||||||||||||||||
| Autjugacy classes | data not computed | ||||||||||||||||
| Dimension | 1 | |
|---|---|---|
| Irr. complex chars. | 2520 | 2520 |
Constructions
| Rank: | $3$ |
| Inequivalent generating triples: | not computed |
Homology
| Primary decomposition: | $C_{2}^{3} \times C_{3}^{2} \times C_{5} \times C_{7}$ |
Subgroups
| Center: | $Z \simeq$ $C_{2} \times C_{6} \times C_{210}$ | $G/Z \simeq$ $C_1$ | |
| Commutator: | $G' \simeq$ $C_1$ | $G/G' \simeq$ $C_{2} \times C_{6} \times C_{210}$ | |
| Frattini: | $\Phi \simeq$ $C_1$ | $G/\Phi \simeq$ $C_{2} \times C_{6} \times C_{210}$ | |
| Fitting: | $\operatorname{Fit} \simeq$ $C_{2} \times C_{6} \times C_{210}$ | $G/\operatorname{Fit} \simeq$ $C_1$ | |
| Radical: | $R \simeq$ $C_{2} \times C_{6} \times C_{210}$ | $G/R \simeq$ $C_1$ | |
| Socle: | $S \simeq$ $C_{2} \times C_{6} \times C_{210}$ | $G/S \simeq$ $C_1$ | |
| 2-Sylow subgroup: | $P_{2} \simeq$ $C_2^3$ | ||
| 3-Sylow subgroup: | $P_{3} \simeq$ $C_3^2$ | ||
| 5-Sylow subgroup: | $P_{5} \simeq$ $C_5$ | ||
| 7-Sylow subgroup: | $P_{7} \simeq$ $C_7$ |