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×#36594 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C7 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C7 |
| prime radical | C2×C7×#127 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C7 ◁ G | |
2330 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)(12,13,16,14)(15,17,19,18)(21,25)(22,27);(12,13,16,14)(15,18,19,17)(20,21)(22,26)(23,27)(24,25);(8,9,10,11)(13,14)(17,18)(20,22,24,27)(21,26,25,23);(12,14,16,13)(15,17,19,18)(20,23,24,26)(21,27,25,22);(12,15,16,19)(13,17,14,18)490 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 29582);