Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2883 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C7 · Schur multiplier (H2) = C210 |
| direct product | C7×#60070 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | C22×C22⋊C4 |
| commutator | C22 | C24×C4×C7 |
| Frattini, cosocle | C23 | C25×C7 |
| socle | C2×C7 | C23×C22⋊C4 |
| prime radical | C2×C7×#585 | C2 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C22×C7×#41 ◁ G | |
4892 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 5 generators on 39 points:
(1,2,3,4,5,6,7)(8,9)(10,18,21,14)(11,15)(12,16)(13,17)(19,30,32,25)(20,31,34,26)(22,27)(23,28)(24,29)(33,38,39,36)(35,37);(8,10)(9,14)(11,19)(12,20)(13,21)(15,25)(16,26)(17,18)(22,32)(23,33)(24,34)(27,30)(28,36)(29,31)(35,39)(37,38);(8,11)(9,15)(10,19)(12,23)(13,22)(14,25)(16,28)(17,27)(18,30)(20,33)(21,32)(24,35)(26,36)(29,37)(31,38)(34,39);(11,22)(15,27)(19,32)(23,35)(25,30)(28,37)(33,39)(36,38);(8,12,13,24)(9,16,17,29)(10,20,21,34)(11,23,22,35)(14,26,18,31)(15,28,27,37)(19,33,32,39)(25,36,30,38)574 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 53058);