This group is not stored in the database. However, basic information about the group, computed on the fly, is listed below.
Group information
| Description: | $C_{2}^{2} \times C_{5060}$ |
| Order: | \(20240\)\(\medspace = 2^{4} \cdot 5 \cdot 11 \cdot 23 \) |
| Exponent: | \(5060\)\(\medspace = 2^{2} \cdot 5 \cdot 11 \cdot 23 \) |
| Automorphism group: | Group of order \(168960\)\(\medspace = 2^{10} \cdot 3 \cdot 5 \cdot 11 \) |
| Outer automorphisms: | Group of order \(168960\)\(\medspace = 2^{10} \cdot 3 \cdot 5 \cdot 11 \) |
| 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 | 4 | 5 | 10 | 11 | 20 | 22 | 23 | 44 | 46 | 55 | 92 | 110 | 115 | 220 | 230 | 253 | 460 | 506 | 1012 | 1265 | 2530 | 5060 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Elements | 1 | 7 | 8 | 4 | 28 | 10 | 32 | 70 | 22 | 80 | 154 | 40 | 176 | 280 | 88 | 320 | 616 | 220 | 704 | 1540 | 1760 | 880 | 6160 | 7040 | 20240 |
| Conjugacy classes | 1 | 7 | 8 | 4 | 28 | 10 | 32 | 70 | 22 | 80 | 154 | 40 | 176 | 280 | 88 | 320 | 616 | 220 | 704 | 1540 | 1760 | 880 | 6160 | 7040 | 20240 |
| Divisions | data not computed | ||||||||||||||||||||||||
| Autjugacy classes | data not computed | ||||||||||||||||||||||||
| Dimension | 1 | |
|---|---|---|
| Irr. complex chars. | 20240 | 20240 |
Constructions
| Rank: | $3$ |
| Inequivalent generating triples: | not computed |
Homology
| Primary decomposition: | $C_{2}^{2} \times C_{4} \times C_{5} \times C_{11} \times C_{23}$ |
Subgroups
| Center: | $Z \simeq$ $C_{2}^{2} \times C_{5060}$ | $G/Z \simeq$ $C_1$ | |
| Commutator: | $G' \simeq$ $C_1$ | $G/G' \simeq$ $C_{2}^{2} \times C_{5060}$ | |
| Frattini: | $\Phi \simeq$ $C_2$ | $G/\Phi \simeq$ $C_{2} \times C_{5060}$ | |
| Fitting: | $\operatorname{Fit} \simeq$ $C_{2}^{2} \times C_{5060}$ | $G/\operatorname{Fit} \simeq$ $C_1$ | |
| Radical: | $R \simeq$ $C_{2}^{2} \times C_{5060}$ | $G/R \simeq$ $C_1$ | |
| Socle: | $S \simeq$ $C_{2} \times C_{5060}$ | $G/S \simeq$ $C_2$ | |
| 2-Sylow subgroup: | $P_{2} \simeq$ $C_2^2\times C_4$ | ||
| 5-Sylow subgroup: | $P_{5} \simeq$ $C_5$ | ||
| 11-Sylow subgroup: | $P_{11} \simeq$ $C_{11}$ | ||
| 23-Sylow subgroup: | $P_{23} \simeq$ $C_{23}$ |