This group is not stored in the database. However, basic information about the group, computed on the fly, is listed below.
Group information
| Description: | $C_{2}^{3} \times C_{528}$ | |
| Order: | \(4224\)\(\medspace = 2^{7} \cdot 3 \cdot 11 \) | |
| Exponent: | \(528\)\(\medspace = 2^{4} \cdot 3 \cdot 11 \) | |
| Automorphism group: | Group of order 1720320 | |
| 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 | 11 | 12 | 16 | 22 | 24 | 33 | 44 | 48 | 66 | 88 | 132 | 176 | 264 | 528 | ||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Elements | 1 | 15 | 2 | 16 | 30 | 32 | 10 | 32 | 64 | 150 | 64 | 20 | 160 | 128 | 300 | 320 | 320 | 640 | 640 | 1280 | 4224 | |
| Conjugacy classes | 1 | 15 | 2 | 16 | 30 | 32 | 10 | 32 | 64 | 150 | 64 | 20 | 160 | 128 | 300 | 320 | 320 | 640 | 640 | 1280 | 4224 | |
| Divisions | data not computed | |||||||||||||||||||||
| Autjugacy classes | data not computed | |||||||||||||||||||||
| Dimension | 1 | |
|---|---|---|
| Irr. complex chars. | 4224 | 4224 |
Constructions
| Rank: | $4$ |
| Inequivalent generating quadruples: | not computed |
Homology
| Primary decomposition: | $C_{2}^{3} \times C_{16} \times C_{3} \times C_{11}$ |
Subgroups
| Center: | $Z \simeq$ $C_{2}^{3} \times C_{528}$ | $G/Z \simeq$ $C_1$ | |
| Commutator: | $G' \simeq$ $C_1$ | $G/G' \simeq$ $C_{2}^{3} \times C_{528}$ | |
| Frattini: | $\Phi \simeq$ $C_8$ | $G/\Phi \simeq$ $C_2^3\times C_{66}$ | |
| Fitting: | $\operatorname{Fit} \simeq$ $C_{2}^{3} \times C_{528}$ | $G/\operatorname{Fit} \simeq$ $C_1$ | |
| Radical: | $R \simeq$ $C_{2}^{3} \times C_{528}$ | $G/R \simeq$ $C_1$ | |
| Socle: | $S \simeq$ $C_2^3\times C_{66}$ | $G/S \simeq$ $C_8$ | |
| 2-Sylow subgroup: | $P_{2} \simeq$ $C_2^3\times C_{16}$ | ||
| 3-Sylow subgroup: | $P_{3} \simeq$ $C_3$ | ||
| 11-Sylow subgroup: | $P_{11} \simeq$ $C_{11}$ |