Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60372 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C28 |
| direct product | C3×#60598 |
| 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×C4×#41 ◁ G | |
2010 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)(4,5)(6,9)(7,8)(10,11)(12,13)(14,19)(15,21)(16,22)(17,31)(18,23)(20,27)(24,33)(25,34)(26,36)(28,37)(29,35)(30,38)(32,40)(39,42)(41,43);(5,8)(9,11)(12,14,18,26)(13,19,23,36)(15,24,28,41)(16,29,17,25)(20,32,38,42)(21,33,37,43)(22,35,31,34)(27,40,30,39);(12,15)(13,20)(14,24)(16,27)(17,30)(18,28)(19,32)(21,31)(22,37)(23,38)(25,39)(26,41)(29,40)(33,34)(35,43)(36,42);(4,6,7,10)(5,9,8,11)(12,15)(13,21)(14,24)(16,30)(17,27)(18,28)(19,33)(20,31)(22,38)(23,37)(25,40)(26,41)(29,39)(32,34)(35,42)(36,43);(12,16,18,17)(13,22,23,31)(14,25,26,29)(15,27,28,30)(19,34,36,35)(20,37,38,21)(24,39,41,40)(32,33,42,43)183 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 53586);