Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C28 |
| direct product | C3×#60682 |
| 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 | |
1878 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)(14,19)(15,27)(16,21)(17,31)(18,22)(20,29)(23,36)(24,33)(25,34)(26,40)(28,35)(30,41)(32,39)(37,43)(38,42);(5,8)(9,11)(12,14)(13,19)(15,23)(16,25)(17,26)(18,24)(20,32)(21,34)(22,33)(27,36)(28,38)(29,39)(30,37)(31,40)(35,42)(41,43);(12,15,17,29,18,30,16,28)(13,20,21,27,22,35,31,41)(14,23,26,39,24,37,25,38)(19,32,34,36,33,42,40,43);(4,6,7,10)(5,9,8,11)(14,24)(19,33)(23,37)(25,26)(32,42)(34,40)(36,43)(38,39);(12,16,18,17)(13,21,22,31)(14,25,24,26)(15,28,30,29)(19,34,33,40)(20,27,35,41)(23,38,37,39)(32,36,42,43)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 53670);