Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C210 |
| direct product | C22×C3×#450 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C3 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C3 |
| prime radical | C24×C3×D4 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24×C3 ◁ C24×C3×C4 ◁ G | |
7126 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 19 points:
(1,2)(17,18,19);(3,4);(5,6)(7,8)(9,10,12,15)(11,13,16,14);(10,13)(11,16)(14,15);(6,8)228 conjugacy classes — view the full character table.
| Fitting subgroup of | C22×D3×#450, C22×#80146 |
| maximal subgroup of | C22×D3×#450, C22×#80146, C22×C3×#3003 |
| maximal normal subgroup of | C22×D3×#450, C22×#80146, C22×C3×#3003 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 53336);