Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2032 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×#11280 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | #393 |
| commutator | C24 | C22×C4×C7 |
| Frattini, cosocle | C23×C4 | C23×C7 |
| socle | C2×C7 | #1442 |
| prime radical | C24×C7 | C22×C4 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C23×C4×C7 ◁ G | |
580 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3,4,5,6,7)(8,9,12,17,13,16,27,33)(10,18,21,34,20,14,26,29)(11,22,25,36,24,15,30,32)(19,28,35,38,23,31,37,39);(8,10)(9,14)(11,23,24,19)(12,26)(13,20)(15,31,22,28)(16,18)(17,34)(21,27)(25,35,30,37)(29,33)(32,38,36,39);(8,11)(9,15,16,22)(10,19)(12,25)(13,24)(14,28,18,31)(17,32,33,36)(20,23)(21,35)(26,37)(27,30)(29,38,34,39)322 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 4268);