Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2104 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×#11318 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | #401 |
| commutator | C24 | C22×C4×C7 |
| Frattini, cosocle | C25 | C23×C7 |
| prime radical | C25×C7 | C23 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C25×C7 ◁ G | |
782 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3,4,5,6,7)(12,13)(14,17,21,25)(15,22)(16,19)(18,23)(20,24,27,26);(8,9)(10,11)(12,14)(13,17,19,25)(15,20,23,27)(16,21)(18,26)(22,24);(9,11)(12,15)(13,18,19,22)(14,20,21,27)(16,23)(17,24)(25,26)322 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 4306);