Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3198 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C28 |
| direct product | C2×C3×#3200 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C3 |
| prime radical | C2×C3×#535 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C22×C3×#41 ◁ G | |
2314 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)(5,9)(6,8)(7,10)(11,12,15,19)(13,16,20,24)(14,17,22,18)(21,23,26,25);(3,5)(4,7)(6,10)(8,9)(11,13,15,20)(12,16,19,24)(14,21,22,26)(17,23,18,25);(5,10)(7,9)(12,17)(14,22)(16,23)(18,19)(21,26)(24,25);(4,8)(7,9)(13,20)(16,24)(21,26)(23,25)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 53732);