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×#60600 |
| 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)(12,13,18,16)(14,17,21,19)(15,22,25,27)(20,26,23,24);(5,8)(9,11)(12,14)(13,19)(15,22)(16,17)(18,21)(20,26)(23,24)(25,27);(12,15,18,25)(13,20,16,23)(14,22,21,27)(17,24,19,26);(4,6,7,10)(5,9,8,11)(12,16,18,13)(14,17,21,19)(15,23,25,20)(22,24,27,26);(12,17,18,19)(13,21,16,14)(15,24,25,26)(20,27,23,22)183 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 53588);