This group is not stored in the database. However, basic information about the group, computed on the fly, is listed below.
Group information
| Description: | $C_{2}^{3} \times C_{160}$ |
| Order: | \(1280\)\(\medspace = 2^{8} \cdot 5 \) |
| Exponent: | \(160\)\(\medspace = 2^{5} \cdot 5 \) |
| Automorphism group: | Group of order \(688128\)\(\medspace = 2^{15} \cdot 3 \cdot 7 \) |
| Outer automorphisms: | Group of order \(688128\)\(\medspace = 2^{15} \cdot 3 \cdot 7 \) |
| 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 | 16 | 20 | 32 | 40 | 80 | 160 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Elements | 1 | 15 | 16 | 4 | 32 | 60 | 64 | 64 | 128 | 128 | 256 | 512 | 1280 |
| Conjugacy classes | 1 | 15 | 16 | 4 | 32 | 60 | 64 | 64 | 128 | 128 | 256 | 512 | 1280 |
| Divisions | data not computed | ||||||||||||
| Autjugacy classes | data not computed | ||||||||||||
| Dimension | 1 | |
|---|---|---|
| Irr. complex chars. | 1280 | 1280 |
Constructions
| Rank: | $4$ |
| Inequivalent generating quadruples: | not computed |
Homology
| Primary decomposition: | $C_{2}^{3} \times C_{32} \times C_{5}$ |
Subgroups
| Center: | $Z \simeq$ $C_{2}^{3} \times C_{160}$ | $G/Z \simeq$ $C_1$ | |
| Commutator: | $G' \simeq$ $C_1$ | $G/G' \simeq$ $C_{2}^{3} \times C_{160}$ | |
| Frattini: | $\Phi \simeq$ $C_{16}$ | $G/\Phi \simeq$ $C_2^3\times C_{10}$ | |
| Fitting: | $\operatorname{Fit} \simeq$ $C_{2}^{3} \times C_{160}$ | $G/\operatorname{Fit} \simeq$ $C_1$ | |
| Radical: | $R \simeq$ $C_{2}^{3} \times C_{160}$ | $G/R \simeq$ $C_1$ | |
| Socle: | $S \simeq$ $C_2^3\times C_{10}$ | $G/S \simeq$ $C_{16}$ | |
| 2-Sylow subgroup: | $P_{2} \simeq$ $C_2^3\times C_{32}$ | ||
| 5-Sylow subgroup: | $P_{5} \simeq$ $C_5$ |