Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33758 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C23×C4 |
| direct product | C3×#33785 |
| 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 | |
594 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)(6,10)(7,9)(8,11)(12,13)(14,24)(15,23)(16,19)(17,30)(18,21)(20,27)(22,32)(25,41)(26,40)(28,35)(29,37)(31,39)(33,38)(34,42)(36,43);(4,6,7,11)(5,8,9,10)(13,19)(20,34)(21,23)(22,35)(24,40)(30,37)(36,39)(38,41);(6,11)(8,10)(12,14)(13,20)(15,25)(16,26)(17,31)(18,33)(19,34)(21,36)(22,38)(23,39)(24,30)(27,32)(28,42)(29,43)(35,41)(37,40);(5,9)(8,10)(12,15,16,18)(13,21,19,23)(14,25,26,33)(17,28,29,32)(20,36,34,39)(22,37,35,30)(24,38,40,41)(27,31,42,43)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26773);