Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C27 |
| direct product | C3×#37115 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C24×C3 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
1340 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)(4,5,8,7)(6,10,11,9)(15,22)(19,24)(21,26)(25,27)(28,29)(30,36)(31,34)(32,35)(33,38)(37,42)(39,41)(40,43);(13,17)(14,20)(19,24)(21,26)(28,29,32,35)(30,33,38,36)(31,34,39,41)(37,40,43,42);(4,6,8,11)(5,9,7,10)(12,13,16,17)(14,18,20,23)(15,19,22,24)(21,25,26,27)(28,29)(30,33)(31,34)(32,35)(36,38)(37,40)(39,41)(42,43);(12,14)(13,18)(15,21)(16,20)(17,23)(19,25)(22,26)(24,27)(28,30,32,38)(29,33,35,36)(31,37,39,43)(34,40,41,42);(4,7,8,5)(6,10,11,9)(12,15)(13,19)(14,21)(16,22)(17,24)(18,25)(20,26)(23,27)(28,31,32,39)(29,34,35,41)(30,37,38,43)(33,40,36,42)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 30103);