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×#60076 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | C22×C22⋊C4 |
| commutator | C22 | C24×C4×C7 |
| Frattini, cosocle | C2×C4 | 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 | |
4812 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)(8,9,13,19,14,18,12,17)(10,20,22,33,23,15,27,29)(11,16,26,32,24,30,25,31)(21,34,36,39,35,28,37,38);(8,10)(9,15)(11,21)(12,22)(13,27)(14,23)(16,28)(17,29)(18,20)(19,33)(24,35)(25,36)(26,37)(30,34)(31,38)(32,39);(8,11)(9,16)(10,21)(12,25)(13,26)(14,24)(15,28)(17,31)(18,30)(19,32)(20,34)(22,36)(23,35)(27,37)(29,38)(33,39);(11,24)(16,30)(21,35)(25,26)(28,34)(31,32)(36,37)(38,39);(8,12,14,13)(9,17,18,19)(10,22,23,27)(11,25,24,26)(15,29,20,33)(16,31,30,32)(21,36,35,37)(28,38,34,39)574 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 53064);