Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23924 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C27 |
| direct product | C3×#23926 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C2×C3×#535 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C22×C3×#41 ◁ G | |
1888 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)(4,5)(6,7)(12,13,16,20)(14,17,21,25)(15,18,23,19)(22,24,27,26);(5,7)(8,9)(10,11)(12,14,16,21)(13,17,20,25)(15,22,23,27)(18,24,19,26);(9,11)(13,18)(15,23)(17,24)(19,20)(22,27)(25,26);(5,7)(14,21)(17,25)(22,27)(24,26)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 16914);