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×#60586 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C3 |
| prime radical | C3×#3002 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×#117 ◁ G | |
2024 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)(15,21)(16,22)(17,31)(18,23)(20,28)(24,33)(25,34)(26,41)(27,35)(29,36)(30,37)(32,38)(39,42)(40,43);(5,8)(9,11)(12,14,18,27)(13,19,23,35)(15,24,29,39)(16,25,17,26)(20,32,37,43)(21,33,36,42)(22,34,31,41)(28,38,30,40);(12,15)(13,20)(14,24)(16,28)(17,30)(18,29)(19,32)(21,31)(22,36)(23,37)(25,38)(26,40)(27,39)(33,41)(34,42)(35,43);(4,6,7,10)(5,9,8,11)(12,15)(13,21)(14,24)(16,30)(17,28)(18,29)(19,33)(20,31)(22,37)(23,36)(25,40)(26,38)(27,39)(32,41)(34,43)(35,42);(12,16,18,17)(13,22,23,31)(14,25,27,26)(15,28,29,30)(19,34,35,41)(20,36,37,21)(24,38,39,40)(32,42,43,33)183 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 53574);