Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#120 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C7 · Schur multiplier (H2) = C28 |
| direct product | C7×#36581 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C7 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C7 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C7 ◁ G | |
2988 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 5 generators on 27 points:
(1,2,3,4,5,6,7)(9,11)(12,13,16,15)(14,18,19,17)(20,21,25,23)(22,26,24,27);(12,13)(14,18)(15,16)(17,19)(20,21,25,23)(22,27,24,26);(8,9)(10,11)(12,14,16,19)(13,17,15,18)(20,22,25,24)(21,26,23,27);(12,14)(13,18)(15,17)(16,19)(20,23,25,21)(22,26,24,27);(12,15,16,13)(14,18,19,17)(20,24,25,22)(21,27,23,26)490 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 29569);