Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29736 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C22×C42 |
| direct product | C3×#29812 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C24×C3 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C2×C3×#117 ◁ G | |
824 conjugacy classes of subgroups.
| minimal degree | μ(G) = 47 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 47 points:
(1,2,3)(8,9)(10,13)(11,12)(14,15)(16,17,20,24,21,25,33,39)(18,22,28,36,27,35,43,46)(19,29,30,37,31,38,32,23)(26,40,42,45,41,47,44,34);(4,5)(6,7)(9,12)(13,15)(16,18)(17,22)(19,26)(20,28)(21,27)(23,34)(24,36)(25,35)(29,40)(30,42)(31,41)(32,44)(33,43)(37,45)(38,47)(39,46);(5,7)(10,14)(13,15)(16,19,21,31)(17,23,25,37)(18,26,27,41)(20,32,33,30)(22,34,35,45)(24,38,39,29)(28,44,43,42)(36,47,46,40);(8,10,11,14)(9,13,12,15)(18,27)(22,35)(26,41)(28,43)(34,45)(36,46)(40,47)(42,44)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 22800);