Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32932 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C22×C42 |
| direct product | C3×#33045 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C3×#2619 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C2×C3×#117 ◁ G | |
930 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 31 points:
(1,2,3)(4,5)(6,7)(8,9)(10,13)(11,12)(14,15)(17,21)(18,24)(19,26)(22,29)(23,28)(25,30);(9,12)(13,15)(16,17,19,21,20,23,26,28)(18,22,24,27,25,29,30,31);(8,10,11,14)(9,13,12,15)(16,18,19,24,20,25,26,30)(17,22,21,27,23,29,28,31);(5,7)(17,23)(21,28)(22,29)(27,31)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26033);