Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of B(4,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C42 · Schur multiplier (H2) = C23×C4 |
| direct product | C3×#10920 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | #120 |
| commutator | C23 | C2×C3×C42 |
| Frattini, cosocle | C23×C4 | C23×C3 |
| socle | C22×C3 | #385 |
| prime radical | C24×C3 | C22×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C23×C3×C4 ◁ G | |
698 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3)(8,9,12,16,13,17,23,29)(10,14,20,26,19,25,33,37)(11,15,22,28,21,27,34,38)(18,30,32,39,31,24,35,36);(4,5,6,7)(8,10)(9,14)(11,18)(12,20)(13,19)(15,24)(16,26)(17,25)(21,31)(22,32)(23,33)(27,30)(28,36)(29,37)(34,35)(38,39);(8,11,12,22,13,21,23,34)(9,15,16,28,17,27,29,38)(10,18,20,32,19,31,33,35)(14,24,26,36,25,30,37,39)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 3908);