Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61629 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C28 |
| direct product | C3×#61666 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C3 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×#532 ◁ G | |
2064 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)(6,10)(7,9)(8,11)(12,13)(14,21)(15,28)(16,24)(17,23)(18,22)(19,27)(20,29)(25,39)(26,34)(30,38)(31,37)(32,36)(33,40)(35,42)(41,43);(4,6)(5,8)(7,11)(9,10)(12,14,17,27)(13,19,23,21)(15,25,30,41)(16,31,18,26)(20,33,36,35)(22,37,24,34)(28,43,38,39)(29,42,32,40);(12,15,17,30)(13,20,23,36)(14,25,27,41)(16,29,18,32)(19,33,21,35)(22,28,24,38)(26,40,31,42)(34,39,37,43);(6,11)(8,10)(12,14)(13,21)(15,25)(16,31)(17,27)(18,26)(19,23)(20,35)(22,34)(24,37)(28,39)(29,42)(30,41)(32,40)(33,36)(38,43);(5,9)(8,10)(12,16,17,18)(13,22,23,24)(14,26,27,31)(15,29,30,32)(19,34,21,37)(20,28,36,38)(25,40,41,42)(33,39,35,43)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 54654);