This group is not stored in the database. However, basic information about the group, computed on the fly, is listed below.
Group information
| Description: | $C_{2} \times C_{4} \times C_{392}$ | |
| Order: | \(3136\)\(\medspace = 2^{6} \cdot 7^{2} \) | |
| Exponent: | \(392\)\(\medspace = 2^{3} \cdot 7^{2} \) | |
| Automorphism group: | Group of order 86016 | |
| 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 | 7 | 8 | 14 | 28 | 49 | 56 | 98 | 196 | 392 | ||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Elements | 1 | 7 | 24 | 6 | 32 | 42 | 144 | 42 | 192 | 294 | 1008 | 1344 | 3136 | |
| Conjugacy classes | 1 | 7 | 24 | 6 | 32 | 42 | 144 | 42 | 192 | 294 | 1008 | 1344 | 3136 | |
| Divisions | data not computed | |||||||||||||
| Autjugacy classes | data not computed | |||||||||||||
| Dimension | 1 | |
|---|---|---|
| Irr. complex chars. | 3136 | 3136 |
Constructions
| Rank: | $3$ |
| Inequivalent generating triples: | not computed |
Homology
| Primary decomposition: | $C_{2} \times C_{4} \times C_{8} \times C_{49}$ |
Subgroups
| Center: | $Z \simeq$ $C_{2} \times C_{4} \times C_{392}$ | $G/Z \simeq$ $C_1$ | |
| Commutator: | $G' \simeq$ $C_1$ | $G/G' \simeq$ $C_{2} \times C_{4} \times C_{392}$ | |
| Frattini: | $\Phi \simeq$ $C_2\times C_{28}$ | $G/\Phi \simeq$ $C_2^2\times C_{14}$ | |
| Fitting: | $\operatorname{Fit} \simeq$ $C_{2} \times C_{4} \times C_{392}$ | $G/\operatorname{Fit} \simeq$ $C_1$ | |
| Radical: | $R \simeq$ $C_{2} \times C_{4} \times C_{392}$ | $G/R \simeq$ $C_1$ | |
| Socle: | $S \simeq$ $C_2^2\times C_{14}$ | $G/S \simeq$ $C_2\times C_{28}$ | |
| 2-Sylow subgroup: | $P_{2} \simeq$ $C_2\times C_4\times C_8$ | ||
| 7-Sylow subgroup: | $P_{7} \simeq$ $C_{49}$ |