Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33607 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24×C4 |
| direct product | C3×#33722 |
| 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 ◁ C3×#117 ◁ C3×#405 ◁ G | |
826 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)(9,14)(11,22)(12,23)(15,24)(16,25)(17,29)(18,28)(20,35)(21,34)(26,27)(30,33)(31,36)(32,39)(37,38);(8,9,15,29,12,17,11,14,13,18,22,27,23,26,24,28)(10,16,30,39,21,32,20,25,19,31,35,38,34,37,33,36);(4,5,6,7)(8,10)(9,16)(11,20)(12,21)(13,19)(14,25)(15,30)(17,32)(18,31)(22,35)(23,34)(24,33)(26,37)(27,38)(28,36)(29,39);(10,19)(16,31)(20,33)(21,34)(25,36)(30,35)(32,37)(38,39)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26710);