Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33605 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24×C4 |
| direct product | C3×#33719 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C3 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ C2×C3×C4×C8 ◁ G | |
834 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)(5,7)(8,9)(10,18)(11,23)(12,15)(13,26)(14,21)(16,19)(17,24)(20,30)(22,37)(25,38)(27,35)(28,39)(29,34)(31,33)(32,36);(8,10,19,33,13,22,25,35,12,20,34,39,24,36,11,21)(9,14,23,32,17,28,29,30,15,27,38,37,26,31,16,18);(4,5,6,7)(9,15)(14,27)(16,29)(17,26)(18,30)(23,38)(28,31)(32,37);(10,20)(14,27)(18,30)(21,35)(22,36)(28,31)(32,37)(33,39)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26707);