Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19745 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C23×C8 |
| direct product | C3×#19745 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#127 |
| commutator, Frattini, cosocle | C42 | C24×C3 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
994 conjugacy classes of subgroups; only the 254 normal subgroups are shown.
This group has 994 conjugacy classes of subgroups — too many to draw. Only its 254 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2,3)(5,8)(6,10)(9,11)(13,16)(14,18)(17,19);(4,5,6,8,7,9,10,11);(12,13,14,16,15,17,18,19);(5,9)(8,11)(13,17)(16,19)120 conjugacy classes — view the full character table.
| maximal subgroup of | C32×#19745, D3×#19745, C3×#489347, … |
| maximal normal subgroup of | C32×#19745, D3×#19745, C3×#489347, … |
| maximal quotient of | C32×#19745, C3×#212918, C9×#19745, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 12733);