Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2002 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C7 · Schur multiplier (H2) = C27 |
| direct product | C7×#23399 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | #120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C7 |
| socle | C22×C7 | C2×#120 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C23×C4×C7 ◁ G | |
2062 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3,4,5,6,7)(8,9)(10,11)(12,13)(14,19)(15,20)(16,24)(17,21)(18,22)(23,26)(25,27);(12,14)(13,18)(15,22)(16,25)(17,23)(19,24)(20,26)(21,27);(9,11)(12,14)(13,19)(15,22)(16,25)(17,23)(18,20)(21,26)(24,27);(12,15,17,16)(13,20,21,24)(14,22,23,25)(18,26,27,19)364 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 16387);