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×#41432 |
| 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 | |
1446 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)(14,15,20,23)(16,19,24,25)(21,22,26,27)(28,29)(30,35)(31,32)(33,34)(36,41)(37,43)(38,40)(39,42);(4,5,7,9)(6,8,10,11)(13,18)(14,20)(19,25)(21,26)(29,34)(32,36)(35,42)(38,43);(4,6,7,10)(5,8,9,11)(12,14)(13,15)(16,21)(17,20)(18,23)(19,22)(24,26)(25,27)(28,30)(29,35)(31,40)(32,38)(33,39)(34,42)(36,43)(37,41);(12,15)(13,14)(16,22)(17,23)(18,20)(19,21)(24,27)(25,26)(28,31)(29,36)(30,37)(32,34)(33,41)(35,38)(39,40)(42,43);(12,16,17,24)(13,19,18,25)(14,21,20,26)(15,22,23,27)(28,32,33,36)(29,31,34,41)(30,38,39,43)(35,40,42,37)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 34420);