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 |
| direct product | C3×#33744 |
| 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 | |
520 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 43 points:
(1,2,3)(4,5)(6,9)(7,8)(10,11)(14,22)(15,24)(16,27)(18,29)(20,30)(21,33)(23,34)(25,41)(26,37)(28,40)(31,43)(32,38)(35,36)(39,42);(5,8)(9,11)(12,13)(14,23)(15,20)(16,21)(17,19)(18,26)(22,34)(24,30)(25,39)(27,33)(28,32)(29,37)(31,35)(36,43)(38,40)(41,42);(12,14,24,36,16,25,28,37,17,26,40,41,27,35,15,22)(13,18,30,42,21,31,32,34,19,23,38,43,33,39,20,29);(4,6,7,10)(5,9,8,11)(13,19)(18,23)(20,32)(21,33)(29,34)(30,38)(31,39)(42,43)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26732);