This group is not stored in the database. However, basic information about the group, computed on the fly, is listed below.
Group information
| Description: | $C_{2}^{3} \times C_{12} \times C_{60}$ | |
| Order: | \(5760\)\(\medspace = 2^{7} \cdot 3^{2} \cdot 5 \) | |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) | |
| Automorphism group: | Group of order 12683575296 | |
| 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 | 10 | 12 | 15 | 20 | 30 | 60 | ||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Elements | 1 | 31 | 8 | 96 | 4 | 248 | 124 | 768 | 32 | 384 | 992 | 3072 | 5760 | |
| Conjugacy classes | 1 | 31 | 8 | 96 | 4 | 248 | 124 | 768 | 32 | 384 | 992 | 3072 | 5760 | |
| Divisions | data not computed | |||||||||||||
| Autjugacy classes | data not computed | |||||||||||||
| Dimension | 1 | |
|---|---|---|
| Irr. complex chars. | 5760 | 5760 |
Constructions
| Rank: | $5$ |
| Inequivalent generating 5-tuples: | not computed |
Homology
| Primary decomposition: | $C_{2}^{3} \times C_{4}^{2} \times C_{3}^{2} \times C_{5}$ |
Subgroups
| Center: | $Z \simeq$ $C_{2}^{3} \times C_{12} \times C_{60}$ | $G/Z \simeq$ $C_1$ | |
| Commutator: | $G' \simeq$ $C_1$ | $G/G' \simeq$ $C_{2}^{3} \times C_{12} \times C_{60}$ | |
| Frattini: | $\Phi \simeq$ $C_2^2$ | $G/\Phi \simeq$ $C_2^3\times C_6\times C_{30}$ | |
| Fitting: | $\operatorname{Fit} \simeq$ $C_{2}^{3} \times C_{12} \times C_{60}$ | $G/\operatorname{Fit} \simeq$ $C_1$ | |
| Radical: | $R \simeq$ $C_{2}^{3} \times C_{12} \times C_{60}$ | $G/R \simeq$ $C_1$ | |
| Socle: | $S \simeq$ $C_2^3\times C_6\times C_{30}$ | $G/S \simeq$ $C_2^2$ | |
| 2-Sylow subgroup: | $P_{2} \simeq$ $C_2^3\times C_4^2$ | ||
| 3-Sylow subgroup: | $P_{3} \simeq$ $C_3^2$ | ||
| 5-Sylow subgroup: | $P_{5} \simeq$ $C_5$ |