This group is not stored in the database. However, basic information about the group, computed on the fly, is listed below.
Group information
| Description: | $C_{2}^{3} \times C_{12}^{2}$ |
| Order: | \(1152\)\(\medspace = 2^{7} \cdot 3^{2} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Automorphism group: | Group of order \(3170893824\)\(\medspace = 2^{24} \cdot 3^{3} \cdot 7 \) |
| Outer automorphisms: | Group of order \(3170893824\)\(\medspace = 2^{24} \cdot 3^{3} \cdot 7 \) |
| 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 | 3 | 4 | 6 | 12 | |
|---|---|---|---|---|---|---|---|
| Elements | 1 | 31 | 8 | 96 | 248 | 768 | 1152 |
| Conjugacy classes | 1 | 31 | 8 | 96 | 248 | 768 | 1152 |
| Divisions | data not computed | ||||||
| Autjugacy classes | data not computed | ||||||
| Dimension | 1 | |
|---|---|---|
| Irr. complex chars. | 1152 | 1152 |
Constructions
| Rank: | $5$ |
| Inequivalent generating 5-tuples: | not computed |
Homology
| Primary decomposition: | $C_{2}^{3} \times C_{4}^{2} \times C_{3}^{2}$ |
Subgroups
| Center: | $Z \simeq$ $C_{2}^{3} \times C_{12}^{2}$ | $G/Z \simeq$ $C_1$ | |
| Commutator: | $G' \simeq$ $C_1$ | $G/G' \simeq$ $C_{2}^{3} \times C_{12}^{2}$ | |
| Frattini: | $\Phi \simeq$ $C_2^2$ | $G/\Phi \simeq$ $C_2^3\times C_6^2$ | |
| Fitting: | $\operatorname{Fit} \simeq$ $C_{2}^{3} \times C_{12}^{2}$ | $G/\operatorname{Fit} \simeq$ $C_1$ | |
| Radical: | $R \simeq$ $C_{2}^{3} \times C_{12}^{2}$ | $G/R \simeq$ $C_1$ | |
| Socle: | $S \simeq$ $C_2^3\times C_6^2$ | $G/S \simeq$ $C_2^2$ | |
| 2-Sylow subgroup: | $P_{2} \simeq$ $C_2^3\times C_4^2$ | ||
| 3-Sylow subgroup: | $P_{3} \simeq$ $C_3^2$ |