Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19755 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#19789 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | D42 |
| commutator, Frattini, cosocle | C42 | C24×C3 |
| prime radical | C3×#1619 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×C42 ◁ G | |
680 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)(5,8)(6,11)(9,10)(12,13,16,20)(14,17,22,26)(15,18,23,19)(21,24,27,25);(4,5,7,10)(6,9,11,8)(12,14,16,22)(13,17,20,26)(15,21,23,27)(18,24,19,25);(13,18)(15,23)(17,24)(19,20)(21,27)(25,26);(5,9)(6,11)(8,10)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 12777);