Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60372 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C28 |
| direct product | C3×#60599 |
| 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×C4×#41 ◁ G | |
2046 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 5 generators on 27 points:
(1,2,3)(4,5)(6,9)(7,8)(10,11)(12,13)(14,21)(15,19)(16,24)(17,20)(18,22)(23,26)(25,27);(5,8)(9,11)(12,13)(14,18)(15,24)(16,19)(17,20)(21,22)(23,27)(25,26);(12,14,17,23)(13,18,20,27)(15,22,16,25)(19,26,24,21);(4,6,7,10)(5,9,8,11)(12,13)(14,18)(15,24)(16,19)(17,20)(21,22)(23,27)(25,26);(12,15,17,16)(13,19,20,24)(14,22,23,25)(18,26,27,21)183 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 53587);