Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1595 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C22×C4 |
| direct product | C3×#9036 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | #120 |
| commutator | C22×C4 | C22×C3×C4 |
| Frattini, cosocle | C2×C42 | C23×C3 |
| socle | C22×C3 | #386 |
| prime radical | C3×#392 | C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C2×C3×C42 ◁ G | |
566 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3)(4,5,9,10,8,6,7,11)(12,13)(14,22,25,26)(15,19,18,24)(16,20)(17,21)(23,27);(4,6,8,5)(7,11,9,10)(12,14,23,24,17,25,16,19)(13,18,27,22,21,15,20,26);(4,7,8,9)(5,10,6,11)(12,15,17,18)(13,19,21,24)(14,20,25,27)(16,22,23,26)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 2024);