Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3055 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C25 |
| direct product | C3×#21164 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C22×D4 |
| commutator | C2×C4 | C23×C3×C4 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| socle | C22×C3 | C2×C4×D4 |
| prime radical | C24×C3 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C23×C3×C8 ◁ G | |
890 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 39 points:
(1,2,3)(4,5)(6,7)(8,9,13,17,14,18,12,27)(10,15,21,29,22,30,20,36)(11,23,26,31,24,37,25,16)(19,32,35,38,33,39,34,28);(5,7)(8,10,13,21,14,22,12,20)(9,15,17,29,18,30,27,36)(11,19,26,35,24,33,25,34)(16,28,23,32,31,38,37,39);(8,11,12,25,14,24,13,26)(9,16,17,23,18,31,27,37)(10,19,20,34,22,33,21,35)(15,28,29,32,30,38,36,39);(11,24)(16,31)(19,33)(23,37)(25,26)(28,38)(32,39)(34,35)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 14152);