Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30437 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#30438 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | #2668 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C22×C3×C4 ◁ G | |
874 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)(7,20)(8,10)(9,14)(11,18)(12,22)(13,23)(16,30)(17,28)(19,26)(21,25)(24,29)(27,31)(32,34)(33,35);(5,10)(7,21)(8,23)(11,26)(12,20)(13,14)(15,28)(16,31)(17,18)(22,24)(30,35)(32,33);(4,6)(5,11)(7,16)(8,17)(9,19)(10,15)(12,27)(13,26)(14,28)(18,23)(20,34)(21,31)(22,33)(24,32)(25,30)(29,35);(4,7,9,22)(5,12,14,20)(6,16,19,33)(8,24,23,21)(10,25,13,29)(11,27,28,34)(15,30,26,35)(17,32,18,31)102 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 23426);