Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23912 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26 |
| direct product | C3×#24229 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C3×#3497 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
1330 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)(5,6)(8,9)(10,14)(11,21)(12,24)(13,17)(15,23)(16,27)(18,35)(19,33)(20,30)(22,34)(25,29)(26,32)(28,37)(31,38)(36,39);(4,5,7,6)(8,10,13,20)(9,14,17,30)(11,18,22,36)(12,25,27,19)(15,31,32,28)(16,33,24,29)(21,35,34,39)(23,38,26,37);(8,11,13,22)(9,15,17,32)(10,18,20,36)(12,26,27,23)(14,28,30,31)(16,34,24,21)(19,38,25,37)(29,39,33,35);(4,6,7,5)(11,22)(15,32)(18,36)(21,34)(23,26)(28,31)(35,39)(37,38)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 17217);