Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1619 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#18522 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | #120 |
| commutator, Frattini, cosocle | C42 | C24×C3 |
| socle | C22×C3 | C2×#120 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C2×C3×C42 ◁ G | |
1000 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)(5,9)(6,7)(10,11)(12,13)(14,23)(15,26)(16,20)(17,21)(18,22)(19,25)(24,27);(4,5,8,10)(6,9,7,11)(12,14,18,15)(13,19,22,27)(16,24,17,25)(20,23,21,26);(4,6,8,7)(5,9,10,11)(12,15,18,14)(13,19,22,27)(16,25,17,24)(20,23,21,26);(12,16,18,17)(13,20,22,21)(14,24,15,25)(19,23,27,26)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 11510);