Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #16249 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C7 · Schur multiplier (H2) = C25×C4 |
| direct product | C7×#16510 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C7 | C24 |
| commutator, Frattini, cosocle | C24 | C24×C7 |
| prime radical | C23×C7×D4 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24×C7 ◁ G | |
2026 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3,4,5,6,7)(9,11)(12,13,14,15)(20,21,24,25)(22,26,27,23);(16,17,18,19)(21,25)(22,27);(12,13)(14,15)(16,17)(18,19)(20,22)(21,26)(23,25)(24,27);(8,9)(10,11)(12,13)(14,15)(20,23,24,26)(21,27,25,22)364 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 9498);