Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C7 · Schur multiplier (H2) = C29 |
| direct product | C7×#60161 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | C22×D4 |
| commutator | C4 | C24×C4×C7 |
| Frattini, cosocle | C2×C4 | C25×C7 |
| socle | C22×C7 | C23×D4 |
| prime radical | C2×C7×#575 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C2×C7×#117 ◁ G | |
2706 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 5 generators on 43 points:
(1,2,3,4,5,6,7)(8,9,11,10)(12,13)(14,23)(15,20)(16,21)(17,30)(18,22)(19,25)(24,37)(26,33)(27,35)(28,36)(29,40)(31,34)(32,39)(38,43)(41,42);(12,14)(13,19)(15,24)(16,25)(17,31)(18,26)(20,32)(21,33)(22,34)(23,30)(27,38)(28,39)(29,41)(35,42)(36,43)(37,40);(8,10,11,9)(12,15)(13,20)(14,24)(16,28)(17,29)(18,27)(19,32)(21,36)(22,35)(23,37)(25,39)(26,38)(30,40)(31,41)(33,43)(34,42);(15,27)(20,35)(24,38)(28,29)(32,42)(36,40)(37,43)(39,41);(12,16,18,17)(13,21,22,30)(14,25,26,31)(15,28,27,29)(19,33,34,23)(20,36,35,40)(24,39,38,41)(32,43,42,37)616 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 53149);