Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1595 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C23 |
| direct product | C7×#9048 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | #120 |
| commutator | C22×C4 | C22×C4×C7 |
| Frattini, cosocle | C2×C42 | C23×C7 |
| socle | C22×C7 | #386 |
| prime radical | C23×C7 | C4×D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C2×C42×C7 ◁ G | |
424 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3,4,5,6,7)(8,9,12,14,13,17,11,16)(10,18,20,15,21,22,19,23)(24,25,28,32,29,26,30,34)(27,33,36,39,35,31,37,38);(9,14)(11,12)(15,18)(16,17)(19,20)(22,23)(24,26,28,34,29,25,30,32)(27,33,37,38,35,31,36,39);(8,10,13,21)(9,15,17,23)(11,20,12,19)(14,18,16,22)(24,27)(25,31)(26,33)(28,37)(29,35)(30,36)(32,39)(34,38)322 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 2036);