Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32894 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C28 |
| direct product | C3×#32895 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | #2846 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C22×C3×C4 ◁ G | |
3540 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)(5,11)(6,15)(8,9)(13,28)(14,20)(16,30)(17,22)(18,21)(23,26)(24,29)(25,34)(31,32);(4,5)(6,12)(7,13)(8,20)(9,11)(10,14)(15,26)(16,25)(17,27)(18,24)(19,29)(21,28)(22,23)(30,35)(31,33)(32,34);(4,6,10,17)(5,12,14,27)(7,16,19,32)(8,15,9,22)(11,23,20,26)(13,25,29,34)(18,31,21,30)(24,33,28,35);(4,7)(5,13)(6,16)(8,21)(9,18)(10,19)(11,24)(12,25)(14,29)(15,30)(17,32)(20,28)(22,31)(23,33)(26,35)(27,34)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 25883);