This group is not stored in the database. However, basic information about the group, computed on the fly, is listed below.
Group information
| Description: | $C_{2}^{2} \times C_{384}$ |
| Order: | \(1536\)\(\medspace = 2^{9} \cdot 3 \) |
| Exponent: | \(384\)\(\medspace = 2^{7} \cdot 3 \) |
| Automorphism group: | Group of order \(12288\)\(\medspace = 2^{12} \cdot 3 \) |
| Outer automorphisms: | Group of order \(12288\)\(\medspace = 2^{12} \cdot 3 \) |
| 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 | 3 | 4 | 6 | 8 | 12 | 16 | 24 | 32 | 48 | 64 | 96 | 128 | 192 | 384 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Elements | 1 | 7 | 2 | 8 | 14 | 16 | 16 | 32 | 32 | 64 | 64 | 128 | 128 | 256 | 256 | 512 | 1536 |
| Conjugacy classes | 1 | 7 | 2 | 8 | 14 | 16 | 16 | 32 | 32 | 64 | 64 | 128 | 128 | 256 | 256 | 512 | 1536 |
| Divisions | data not computed | ||||||||||||||||
| Autjugacy classes | data not computed | ||||||||||||||||
| Dimension | 1 | |
|---|---|---|
| Irr. complex chars. | 1536 | 1536 |
Constructions
| Rank: | $3$ |
| Inequivalent generating triples: | not computed |
Homology
| Primary decomposition: | $C_{2}^{2} \times C_{128} \times C_{3}$ |
Subgroups
| Center: | $Z \simeq$ $C_{2}^{2} \times C_{384}$ | $G/Z \simeq$ $C_1$ | |
| Commutator: | $G' \simeq$ $C_1$ | $G/G' \simeq$ $C_{2}^{2} \times C_{384}$ | |
| Frattini: | $\Phi \simeq$ $C_{64}$ | $G/\Phi \simeq$ $C_2^2\times C_6$ | |
| Fitting: | $\operatorname{Fit} \simeq$ $C_{2}^{2} \times C_{384}$ | $G/\operatorname{Fit} \simeq$ $C_1$ | |
| Radical: | $R \simeq$ $C_{2}^{2} \times C_{384}$ | $G/R \simeq$ $C_1$ | |
| Socle: | $S \simeq$ $C_2^2\times C_6$ | $G/S \simeq$ $C_{64}$ | |
| 2-Sylow subgroup: | $P_{2} \simeq$ $C_{2}^{2} \times C_{128}$ | ||
| 3-Sylow subgroup: | $P_{3} \simeq$ $C_3$ |