Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C25 |
| direct product | C3×#21181 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | #121 |
| commutator | C2×C4 | C23×C3×C4 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| socle | C22×C3 | C2×#121 |
| prime radical | C24×C3 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C23×C3×C4 ◁ G | |
874 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)(4,5,7,6)(8,9)(10,20)(11,25)(12,17)(13,18)(14,19)(15,23)(16,27)(21,35)(22,30)(24,33)(26,32)(28,31)(29,38)(34,37)(36,39);(4,6,7,5)(8,10)(9,15)(11,21)(12,22)(13,23)(14,24)(16,29)(17,20)(18,33)(19,30)(25,34)(26,36)(27,37)(28,38)(31,39)(32,35);(8,11,13,27,14,28,12,26)(9,16,17,25,19,32,18,31)(10,21,23,37,24,38,22,36)(15,29,20,34,30,35,33,39);(8,12,14,13)(9,17,19,18)(10,22,24,23)(11,26,28,27)(15,20,30,33)(16,25,32,31)(21,36,38,37)(29,34,35,39)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 14169);