Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C27 |
| direct product | C3×#41440 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C22×C3×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
1406 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 5 generators on 43 points:
(1,2,3)(6,10)(8,11)(12,13,17,14)(15,20,23,18)(16,19,24,21)(22,26,27,25)(28,29)(30,35)(31,39)(32,37)(33,34)(36,42)(38,41)(40,43);(4,5,7,9)(6,8,10,11)(13,14)(18,20)(19,21)(25,26)(29,34)(32,35)(36,39)(41,43);(4,6,7,10)(5,8,9,11)(12,14,17,13)(15,20,23,18)(16,21,24,19)(22,26,27,25)(28,30,33,37)(29,35,34,32)(31,40,42,38)(36,41,39,43);(12,15,17,23)(13,18,14,20)(16,22,24,27)(19,25,21,26)(28,31,33,42)(29,36,34,39)(30,38,37,40)(32,41,35,43);(12,16,17,24)(13,19,14,21)(15,22,23,27)(18,25,20,26)(28,32,33,35)(29,37,34,30)(31,41,42,43)(36,40,39,38)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 34428);