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×#41434 |
| 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 | |
1438 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)(14,21,19,15)(16,20,24,25)(22,27,26,23)(28,29)(30,37)(31,32)(33,34)(35,40)(36,42)(38,39)(41,43);(4,5,7,9)(6,8,10,11)(13,18)(14,19)(20,25)(22,26)(29,34)(32,36)(35,37)(39,43);(4,6,7,10)(5,8,9,11)(12,14,17,19)(13,15,18,21)(16,22,24,26)(20,23,25,27)(28,30,33,40)(29,35,34,37)(31,41,42,38)(32,39,36,43);(12,15)(13,19)(14,18)(16,23)(17,21)(20,26)(22,25)(24,27)(28,31)(29,36)(30,38)(32,34)(33,42)(35,39)(37,43)(40,41);(12,16,17,24)(13,20,18,25)(14,22,19,26)(15,23,21,27)(28,32,33,36)(29,31,34,42)(30,39,40,43)(35,41,37,38)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 34422);