Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C27 |
| direct product | C3×#41400 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C3×#392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
1434 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)(6,10)(8,11)(12,13,17,18)(14,19,21,16)(15,23,25,20)(22,24,27,26)(28,29,33,37)(30,34,38,43)(31,35,41,40)(32,36,42,39);(4,5,7,9)(6,8,10,11)(13,18)(14,21)(20,23)(22,27)(30,38)(32,42)(34,43)(36,39);(4,6)(5,8)(7,10)(9,11)(12,14)(13,19)(15,22)(16,18)(17,21)(20,26)(23,24)(25,27)(28,30)(29,34)(31,36)(32,40)(33,38)(35,42)(37,43)(39,41);(12,15,17,25)(13,20,18,23)(14,22,21,27)(16,24,19,26)(28,31)(29,35)(30,39)(32,34)(33,41)(36,38)(37,40)(42,43);(12,16,17,19)(13,14,18,21)(15,24,25,26)(20,22,23,27)(28,32)(29,36)(30,40)(31,34)(33,42)(35,38)(37,39)(41,43)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 34388);