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×#9038 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | #120 |
| commutator | C22×C4 | C22×C4×C7 |
| Frattini, cosocle | C23×C4 | C23×C7 |
| socle | C22×C7 | #386 |
| prime radical | C7×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C23×C4×C7 ◁ G | |
468 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3,4,5,6,7)(8,9)(10,13)(11,18,19,14)(12,15)(16,20,22,23)(17,21)(25,29,32,35)(26,33,34,36);(8,10,12,17)(9,13,15,21)(11,16,19,22)(14,20,18,23)(24,25,30,29,28,32,27,35)(26,31,33,37,34,39,36,38);(8,11)(9,14)(10,16)(12,19)(13,20)(15,18)(17,22)(21,23)(24,26)(25,31)(27,36)(28,34)(29,37)(30,33)(32,39)(35,38)322 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 2026);