This group is not stored in the database. However, basic information about the group, computed on the fly, is listed below.
Group information
| Description: | $C_{2}^{3} \times C_{12} \times C_{1092}$ | |
| Order: | \(104832\)\(\medspace = 2^{7} \cdot 3^{2} \cdot 7 \cdot 13 \) | |
| Exponent: | \(1092\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \cdot 13 \) | |
| Automorphism group: | Group of order 228304355328 | |
| 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 | 7 | 12 | 13 | 14 | 21 | 26 | 28 | 39 | 42 | 52 | 78 | 84 | 91 | 156 | 182 | 273 | 364 | 546 | 1092 | ||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Elements | 1 | 31 | 8 | 96 | 248 | 6 | 768 | 12 | 186 | 48 | 372 | 576 | 96 | 1488 | 1152 | 2976 | 4608 | 72 | 9216 | 2232 | 576 | 6912 | 17856 | 55296 | 104832 | |
| Conjugacy classes | 1 | 31 | 8 | 96 | 248 | 6 | 768 | 12 | 186 | 48 | 372 | 576 | 96 | 1488 | 1152 | 2976 | 4608 | 72 | 9216 | 2232 | 576 | 6912 | 17856 | 55296 | 104832 | |
| Divisions | data not computed | |||||||||||||||||||||||||
| Autjugacy classes | data not computed | |||||||||||||||||||||||||
| Dimension | 1 | |
|---|---|---|
| Irr. complex chars. | 104832 | 104832 |
Constructions
| Rank: | $5$ |
| Inequivalent generating 5-tuples: | not computed |
Homology
| Primary decomposition: | $C_{2}^{3} \times C_{4}^{2} \times C_{3}^{2} \times C_{7} \times C_{13}$ |
Subgroups
| Center: | $Z \simeq$ $C_{2}^{3} \times C_{12} \times C_{1092}$ | $G/Z \simeq$ $C_1$ | |
| Commutator: | $G' \simeq$ $C_1$ | $G/G' \simeq$ $C_{2}^{3} \times C_{12} \times C_{1092}$ | |
| Frattini: | $\Phi \simeq$ $C_2^2$ | $G/\Phi \simeq$ $C_{2}^{2} \times C_{6552}$ | |
| Fitting: | $\operatorname{Fit} \simeq$ $C_{2}^{3} \times C_{12} \times C_{1092}$ | $G/\operatorname{Fit} \simeq$ $C_1$ | |
| Radical: | $R \simeq$ $C_{2}^{3} \times C_{12} \times C_{1092}$ | $G/R \simeq$ $C_1$ | |
| Socle: | $S \simeq$ $C_{2}^{2} \times C_{6552}$ | $G/S \simeq$ $C_2^2$ | |
| 2-Sylow subgroup: | $P_{2} \simeq$ $C_2^3\times C_4^2$ | ||
| 3-Sylow subgroup: | $P_{3} \simeq$ $C_3^2$ | ||
| 7-Sylow subgroup: | $P_{7} \simeq$ $C_7$ | ||
| 13-Sylow subgroup: | $P_{13} \simeq$ $C_{13}$ |