Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C23 |
| direct product | C3×#29184 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C3 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C3×C43 ◁ G | |
642 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 768 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(4,5)(6,10)(7,8)(9,11)(12,13,15,17)(14,16,18,19)(20,21,25,27)(22,23,26,24);(5,8)(9,10)(12,14,15,18)(13,16,17,19);(14,18)(16,19)(20,22,24,21,25,26,23,27);(4,6,7,11)(5,9,8,10)(20,23,25,24)(21,26,27,22)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 22172);