Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2190 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26 |
| direct product | C3×#33530 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | #447 |
| commutator, Frattini, cosocle | C2×C8 | C24×C3 |
| socle | C2×C3 | C2×#447 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C22×C3×C4 ◁ C22×C3×M16 ◁ G | |
1132 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2,3)(4,5)(6,15)(7,12)(8,20)(9,24)(10,14)(11,17)(13,23)(16,32)(18,31)(19,30)(21,26)(22,34)(25,29)(27,33)(28,35);(4,6)(5,11)(7,16)(8,21)(9,25)(10,18)(12,26)(13,29)(14,27)(15,20)(17,23)(19,33)(22,35)(24,28)(30,32)(31,34);(8,22)(13,30)(15,31)(17,33)(19,23)(20,34)(21,35)(29,32);(4,7,10,9)(5,12,14,24)(6,16,18,25)(8,19,22,23)(11,26,27,28)(13,20,30,34)(15,32,31,29)(17,21,33,35)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26518);