Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #14715 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C27×C4 |
| direct product | C3×#15124 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C3 | C24 |
| commutator, Frattini, cosocle | C24 | C24×C3 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24×C3 ◁ G | |
3822 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)(9,10)(17,19);(8,9,11,10)(13,15)(17,19);(12,13)(14,15)(16,17)(18,19);(4,5,6,7)(8,10,11,9)192 conjugacy classes — view the full character table.
| Fitting subgroup of | #72271205, #72271211, D3×#15124, … |
| maximal subgroup of | #72271205, #72271211, C3×#4306523, … |
| maximal normal subgroup of | #72271205, #72271211, C3×#4306523, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 8112);