Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18682 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#18706 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C3 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
908 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2,3)(5,9)(7,14)(10,16)(11,15)(13,18)(17,19)(20,21)(22,30)(23,28)(24,27)(25,29)(26,32)(31,35)(33,34);(4,5,8,11)(6,10,12,17)(7,15,14,9)(13,19,18,16)(20,22,25,33)(21,26,29,35)(23,31,24,32)(27,34,28,30);(4,6)(5,10)(7,13)(8,12)(9,16)(11,17)(14,18)(15,19)(21,27)(23,24)(26,34)(28,29)(30,35)(31,32);(6,12)(10,17)(13,18)(16,19)(20,23,25,24)(21,28,29,27)(22,31,33,32)(26,30,35,34)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 11694);