This group is not stored in the database. However, basic information about the group, computed on the fly, is listed below.
Group information
| Description: | $C_{2}^{2} \times C_{540}$ | |
| Order: | \(2160\)\(\medspace = 2^{4} \cdot 3^{3} \cdot 5 \) | |
| Exponent: | \(540\)\(\medspace = 2^{2} \cdot 3^{3} \cdot 5 \) | |
| Automorphism group: | Group of order 13824 | |
| 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 | 3 | 4 | 5 | 6 | 9 | 10 | 12 | 15 | 18 | 20 | 27 | 30 | 36 | 45 | 54 | 60 | 90 | 108 | 135 | 180 | 270 | 540 | ||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Elements | 1 | 7 | 2 | 8 | 4 | 14 | 6 | 28 | 16 | 8 | 42 | 32 | 18 | 56 | 48 | 24 | 126 | 64 | 168 | 144 | 72 | 192 | 504 | 576 | 2160 | |
| Conjugacy classes | 1 | 7 | 2 | 8 | 4 | 14 | 6 | 28 | 16 | 8 | 42 | 32 | 18 | 56 | 48 | 24 | 126 | 64 | 168 | 144 | 72 | 192 | 504 | 576 | 2160 | |
| Divisions | data not computed | |||||||||||||||||||||||||
| Autjugacy classes | data not computed | |||||||||||||||||||||||||
| Dimension | 1 | |
|---|---|---|
| Irr. complex chars. | 2160 | 2160 |
Constructions
| Rank: | $3$ |
| Inequivalent generating triples: | not computed |
Homology
| Primary decomposition: | $C_{2}^{2} \times C_{4} \times C_{27} \times C_{5}$ |
Subgroups
| Center: | $Z \simeq$ $C_{2}^{2} \times C_{540}$ | $G/Z \simeq$ $C_1$ | |
| Commutator: | $G' \simeq$ $C_1$ | $G/G' \simeq$ $C_{2}^{2} \times C_{540}$ | |
| Frattini: | $\Phi \simeq$ $C_{18}$ | $G/\Phi \simeq$ $C_2^2\times C_{30}$ | |
| Fitting: | $\operatorname{Fit} \simeq$ $C_{2}^{2} \times C_{540}$ | $G/\operatorname{Fit} \simeq$ $C_1$ | |
| Radical: | $R \simeq$ $C_{2}^{2} \times C_{540}$ | $G/R \simeq$ $C_1$ | |
| Socle: | $S \simeq$ $C_2^2\times C_{30}$ | $G/S \simeq$ $C_{18}$ | |
| 2-Sylow subgroup: | $P_{2} \simeq$ $C_2^2\times C_4$ | ||
| 3-Sylow subgroup: | $P_{3} \simeq$ $C_{27}$ | ||
| 5-Sylow subgroup: | $P_{5} \simeq$ $C_5$ |