Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30434 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#30436 |
| 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 | |
882 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)(7,11)(8,12)(9,14)(10,18)(13,23)(16,31)(17,25)(19,29)(20,30)(21,26)(22,28)(24,32)(27,34)(33,35);(4,6)(5,10)(7,20)(8,17)(9,19)(11,24)(12,15)(13,29)(14,25)(16,22)(18,23)(21,33)(26,30)(27,31)(28,35)(32,34);(4,7,9,22)(5,11,14,28)(6,16,19,20)(8,21,23,34)(10,24,25,35)(12,26,13,27)(15,30,29,31)(17,32,18,33);(5,12)(6,17)(8,23)(11,26)(13,14)(15,29)(16,32)(18,19)(20,33)(21,34)(27,28)(30,31)102 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 23424);