Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C28 |
| direct product | C3×#42915 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C22×C3×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
1898 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 5 generators on 39 points:
(1,2,3)(5,7)(24,25,30,29)(26,33,27,31)(28,36,37,32)(34,38,35,39);(9,13)(11,19)(14,16)(17,23)(24,26,30,27)(25,31,29,33)(28,34,37,35)(32,38,36,39);(4,5)(6,7)(8,9)(10,16)(11,15)(12,13)(14,18)(17,22)(19,20)(21,23)(24,27,30,26)(25,31,29,33)(28,34,37,35)(32,39,36,38);(8,10,12,18)(9,14,13,16)(11,17,19,23)(15,21,20,22)(24,28,30,37)(25,32,29,36)(26,34,27,35)(31,38,33,39);(8,11,12,19)(9,15,13,20)(10,17,18,23)(14,21,16,22)(24,29,30,25)(26,31,27,33)(28,36,37,32)(34,38,35,39)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 35903);