Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #14715 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C28 |
| direct product | C3×#15123 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C3 | C24 |
| commutator, Frattini, cosocle | C24 | C24×C3 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24×C3 ◁ G | |
3458 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2,3)(5,7)(13,15)(16,17)(18,19);(9,11)(13,15)(17,19);(8,9)(10,11)(12,13)(14,15)(16,17)(18,19);(4,5,6,7)192 conjugacy classes — view the full character table.
| Fitting subgroup of | D3×#15123, #72270737, #72270809, … |
| maximal subgroup of | D3×#15123, #72270737, C3×#616967, … |
| maximal normal subgroup of | D3×#15123, #72270737, C3×#616967, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 8111);