Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60367 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C28 |
| direct product | C3×#60569 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C3 |
| prime radical | C22×C3×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×#117 ◁ G | |
2010 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 43 points:
(1,2,3)(4,5)(6,9)(7,8)(10,11)(12,13,18,23)(14,19,25,33)(15,28,30,37)(16,22,17,21)(20,31,36,29)(24,38,39,43)(26,35,27,34)(32,41,42,40);(5,8)(9,11)(12,14,18,25)(13,19,23,33)(15,24,30,39)(16,26,17,27)(20,32,36,42)(21,34,22,35)(28,38,37,43)(29,40,31,41);(12,15)(13,20)(14,24)(16,29)(17,31)(18,30)(19,32)(21,28)(22,37)(23,36)(25,39)(26,40)(27,41)(33,42)(34,38)(35,43);(4,6,7,10)(5,9,8,11)(14,25)(19,33)(24,39)(26,27)(32,42)(34,35)(38,43)(40,41);(12,16,18,17)(13,21,23,22)(14,26,25,27)(15,29,30,31)(19,34,33,35)(20,28,36,37)(24,40,39,41)(32,38,42,43)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 53557);