Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C42 · Schur multiplier (H2) = C26×C4 |
| direct product | C22×C3×H(3,ℤ/4) |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C4 | C42 |
| commutator | C4 | C22×C3×C42 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| socle | C23×C3 | #95 |
| prime radical | C25×C3 | D4 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C4 ◁ G | |
2270 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2)(21,22,23);(3,4);(5,6)(7,13,15,19)(8,11)(9,12)(10,14,17,20)(16,18);(5,7)(6,10,12,17)(8,14)(9,15)(11,13,18,19)(16,20)264 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 13325);