This group is not stored in the database. However, basic information about the group, computed on the fly, is listed below.
Group information
| Description: | $C_{2}^{2} \times C_{4} \times C_{40}$ |
| Order: | \(640\)\(\medspace = 2^{7} \cdot 5 \) |
| Exponent: | \(40\)\(\medspace = 2^{3} \cdot 5 \) |
| Automorphism group: | Group of order \(786432\)\(\medspace = 2^{18} \cdot 3 \) |
| Outer automorphisms: | Group of order \(786432\)\(\medspace = 2^{18} \cdot 3 \) |
| 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 | 8 | 10 | 20 | 40 | |
|---|---|---|---|---|---|---|---|---|---|
| Elements | 1 | 15 | 48 | 4 | 64 | 60 | 192 | 256 | 640 |
| Conjugacy classes | 1 | 15 | 48 | 4 | 64 | 60 | 192 | 256 | 640 |
| Divisions | data not computed | ||||||||
| Autjugacy classes | data not computed | ||||||||
| Dimension | 1 | |
|---|---|---|
| Irr. complex chars. | 640 | 640 |
Constructions
| Rank: | $4$ |
| Inequivalent generating quadruples: | not computed |
Homology
| Primary decomposition: | $C_{2}^{2} \times C_{4} \times C_{8} \times C_{5}$ |
Subgroups
| Center: | $Z \simeq$ $C_{2}^{2} \times C_{4} \times C_{40}$ | $G/Z \simeq$ $C_1$ | |
| Commutator: | $G' \simeq$ $C_1$ | $G/G' \simeq$ $C_{2}^{2} \times C_{4} \times C_{40}$ | |
| Frattini: | $\Phi \simeq$ $C_2\times C_4$ | $G/\Phi \simeq$ $C_2^3\times C_{10}$ | |
| Fitting: | $\operatorname{Fit} \simeq$ $C_{2}^{2} \times C_{4} \times C_{40}$ | $G/\operatorname{Fit} \simeq$ $C_1$ | |
| Radical: | $R \simeq$ $C_{2}^{2} \times C_{4} \times C_{40}$ | $G/R \simeq$ $C_1$ | |
| Socle: | $S \simeq$ $C_2^3\times C_{10}$ | $G/S \simeq$ $C_2\times C_4$ | |
| 2-Sylow subgroup: | $P_{2} \simeq$ $C_2^2\times C_4\times C_8$ | ||
| 5-Sylow subgroup: | $P_{5} \simeq$ $C_5$ |