Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32950 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26 |
| direct product | C3×#32951 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | #2847 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C22×C3×C4 ◁ G | |
1298 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)(5,10)(6,15)(8,11)(13,23)(14,24)(16,31)(17,18)(19,20)(22,26)(25,34)(28,29)(30,32);(4,5,11,24,9,14,8,10)(6,12,15,26,18,27,17,22)(7,13,20,28,21,29,19,23)(16,25,30,35,32,34,31,33);(4,6)(5,12)(7,16)(8,17)(9,18)(10,22)(11,15)(13,25)(14,27)(19,31)(20,30)(21,32)(23,33)(24,26)(28,35)(29,34);(4,7)(5,13)(6,16)(8,19)(9,21)(10,23)(11,20)(12,25)(14,29)(15,30)(17,31)(18,32)(22,33)(24,28)(26,35)(27,34)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 25939);