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×#36580 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C7 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C7 |
| prime radical | C7×#2856 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C7 ◁ G | |
2560 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 5 generators on 27 points:
(1,2,3,4,5,6,7)(9,11)(12,13,17,15)(14,18,16,19)(20,21)(22,24)(23,26)(25,27);(12,13,17,15)(14,19,16,18)(20,22,25,26)(21,23,27,24);(8,9)(10,11)(12,14,17,16)(13,18,15,19)(20,23,25,24)(21,26,27,22);(12,15,17,13)(14,18,16,19)(20,21)(22,23)(24,26)(25,27);(12,16,17,14)(13,19,15,18)(20,24,25,23)(21,22,27,26)490 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 29568);