Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32898 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C27 |
| direct product | C3×#32926 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#120 |
| commutator, Frattini, cosocle | C24 | C24×C3 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G | |
1922 conjugacy classes of subgroups; only the 298 normal subgroups are shown.
This group has 1922 conjugacy classes of subgroups — too many to draw. Only its 298 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 15 points:
(1,2,3)(5,7)(12,15);(4,5,6,7);(8,9)(10,12)(11,13)(14,15);(8,10)(9,12)(11,14)(13,15)147 conjugacy classes — view the full character table.
| Fitting subgroup of | D3×#32926, #72270988, #72271060, … |
| maximal subgroup of | D3×#32926, #72270988, C3×D4×B(4,𝔽2), … |
| maximal normal subgroup of | D3×#32926, #72270988, C3×D4×B(4,𝔽2), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 25914);