Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26 |
| direct product | C3×#23539 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C4 | C2×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| socle | C23×C3 | C22×D4 |
| prime radical | C22×C3×#135 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C4 ◁ C22×C3×C42 ◁ G | |
1494 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(4,5)(6,7)(12,13)(14,15)(16,21)(17,22)(18,23)(19,20)(24,26)(25,27);(8,9)(10,11)(12,14)(13,19)(15,21)(16,24)(17,20)(18,25)(22,27)(23,26);(5,7)(9,11)(12,15,16,26,18,27,17,19)(13,20,22,25,23,24,21,14);(12,16,18,17)(13,21,23,22)(14,24,25,20)(15,26,27,19)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 16527);