Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C28 |
| direct product | C2×C3×#3091 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C4 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C3 |
| socle | C23×C3 | C22×D4 |
| prime radical | C2×C3×#525 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C4 ◁ C24×C3×C4 ◁ G | |
2378 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 29 points:
(1,2)(27,28,29);(3,4)(5,8)(6,7)(9,10)(11,12)(13,20)(14,18)(15,23)(16,19)(17,21)(22,25)(24,26);(4,7)(8,10);(3,5,6,9)(4,8,7,10)(11,13)(12,17)(14,21)(15,24)(16,22)(18,25)(19,26)(20,23);(11,14,16,15)(12,18,19,23)(13,21,22,24)(17,25,26,20)228 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 53494);