Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3198 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C27×C4 |
| direct product | C3×#60797 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C3 |
| socle | C22×C3 | C23×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C2×C3×#117 ◁ G | |
2062 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)(14,19)(15,27)(16,21)(17,30)(18,22)(20,28)(23,37)(24,33)(25,34)(26,40)(29,35)(31,36)(32,39)(38,43)(41,42);(4,6,7,11)(5,8,9,10)(12,14)(13,19)(15,23)(16,25)(17,26)(18,24)(20,32)(21,34)(22,33)(27,37)(28,39)(29,38)(30,40)(31,41)(35,43)(36,42);(6,11)(8,10)(12,15)(13,20)(14,23)(16,28)(17,31)(18,29)(19,32)(21,35)(22,36)(24,38)(25,39)(26,41)(27,30)(33,42)(34,43)(37,40);(5,9)(8,10)(14,24)(19,33)(23,38)(25,26)(32,42)(34,40)(37,43)(39,41);(12,16,18,17)(13,21,22,30)(14,25,24,26)(15,28,29,31)(19,34,33,40)(20,35,36,27)(23,39,38,41)(32,43,42,37)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 53785);