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×#61665 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C3 |
| prime radical | C3×#3499 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×#532 ◁ G | |
2008 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)(6,10)(7,9)(8,11)(12,13,18,22)(14,19,25,35)(15,26,28,36)(16,21,17,30)(20,31,37,27)(23,38,41,43)(24,33,29,34)(32,39,42,40);(4,6)(5,8)(7,11)(9,10)(12,14)(13,19)(15,23)(16,29)(17,24)(18,25)(20,32)(21,34)(22,35)(26,38)(27,40)(28,41)(30,33)(31,39)(36,43)(37,42);(12,15,18,28)(13,20,22,37)(14,23,25,41)(16,27,17,31)(19,32,35,42)(21,36,30,26)(24,39,29,40)(33,38,34,43);(6,11)(8,10)(12,14)(13,19)(15,23)(16,29)(17,24)(18,25)(20,32)(21,34)(22,35)(26,38)(27,40)(28,41)(30,33)(31,39)(36,43)(37,42);(5,9)(8,10)(12,16,18,17)(13,21,22,30)(14,24,25,29)(15,27,28,31)(19,33,35,34)(20,36,37,26)(23,39,41,40)(32,38,42,43)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 54653);