Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33522 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#33587 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C3 |
| prime radical | C3×#3155 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×#117 ◁ C3×#405 ◁ G | |
712 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 39 points:
(1,2,3)(5,7)(10,18)(11,21)(12,23)(14,26)(16,31)(17,30)(19,36)(20,35)(22,25)(24,34)(27,38)(28,39)(29,37)(32,33);(4,5,6,7)(8,9)(10,14)(11,16)(12,17)(13,15)(18,26)(19,28)(20,27)(21,31)(22,32)(23,30)(24,29)(25,33)(34,37)(35,38)(36,39);(8,10,13,20)(9,14,15,27)(11,22,24,36)(12,25,23,19)(16,32,29,39)(17,33,30,28)(18,34,35,21)(26,37,38,31);(9,15)(14,27)(16,29)(17,30)(26,38)(28,33)(31,37)(32,39)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26575);