Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19755 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#19759 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | D42 |
| commutator, Frattini, cosocle | C42 | C24×C3 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×C42 ◁ G | |
762 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(4,5)(6,8)(7,10)(9,11)(13,17)(15,23)(18,24)(19,20)(21,27)(25,26);(12,13,16,20)(14,18,22,26)(15,19,23,17)(21,25,27,24);(5,8)(6,11)(9,10)(12,14,16,22)(13,18,20,26)(15,21,23,27)(17,24,19,25);(13,19)(15,23)(17,20)(18,25)(21,27)(24,26)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 12747);