Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18252 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26 |
| direct product | C3×#18261 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C3×#3536 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G | |
1192 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)(8,11)(12,13,18,23)(14,24,15,20)(16,22,17,21)(19,26,27,25);(4,5)(6,8)(7,9)(10,11)(12,14,18,15)(13,19,23,27)(16,25,17,26)(20,21,24,22);(4,6)(5,8)(7,10)(9,11)(12,15,18,14)(13,20,23,24)(16,25,17,26)(19,21,27,22);(12,16,18,17)(13,21,23,22)(14,25,15,26)(19,24,27,20)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 11249);