Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2010 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C26 |
| direct product | C3×#10994 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | #127 |
| commutator | C22×C4 | C22×C3×C4 |
| Frattini, cosocle | C23×C4 | C23×C3 |
| socle | C22×C3 | #392 |
| prime radical | C2×C3×#525 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C23×C3×C4 ◁ G | |
1494 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,3)(4,5,6,7)(8,9)(10,16)(11,19)(12,15)(13,17)(14,21)(18,23)(20,22);(8,10)(9,13)(11,20)(12,18)(14,23)(15,22)(16,19)(17,21);(13,22)(16,23)156 conjugacy classes — view the full character table.
| Fitting subgroup of | #72248356, #72248358, #72248394, … |
| maximal subgroup of | C3×#421068, #72248356, #72248358, … |
| maximal normal subgroup of | C3×#421068, #72248356, #72248358, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 3982);