Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61629 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C28 |
| direct product | C3×#61668 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C3 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×#532 ◁ G | |
2200 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 27 points:
(1,2,3)(4,5)(6,10)(7,9)(8,11)(12,13)(14,19)(15,21)(16,22)(17,26)(18,23)(20,24)(25,27);(4,6)(5,8)(7,11)(9,10)(12,14,18,15)(13,19,23,21)(16,25,17,24)(20,22,27,26);(12,14,18,15)(13,20,23,27)(16,24,17,25)(19,22,21,26);(6,11)(8,10)(12,15,18,14)(13,21,23,19)(16,24,17,25)(20,26,27,22);(5,9)(8,10)(12,16,18,17)(13,22,23,26)(14,24,15,25)(19,20,21,27)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 54656);