Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11505 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C26 |
| direct product | C3×#12694 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#99 |
| commutator | C22×C4 | C22×C3×C4 |
| Frattini, cosocle | C22×D4 | C23×C3 |
| prime radical | C23×C3×D4 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C24×C3 ◁ C23×C3×D4 ◁ G | |
1206 conjugacy classes of subgroups; only the 158 normal subgroups are shown.
This group has 1206 conjugacy classes of subgroups — too many to draw. Only its 158 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,3)(4,5)(6,7)(8,9)(10,16,18,23)(11,19,21,14)(12,15)(13,17)(20,22);(8,10,12,18)(9,13,15,22)(11,20,21,17)(14,23,19,16);(5,7)120 conjugacy classes — view the full character table.
| Fitting subgroup of | #72253549, D3×#12694, #72255216, … |
| maximal subgroup of | C3×#493370, #72253549, D3×#12694, … |
| maximal normal subgroup of | C3×#493370, #72253549, D3×#12694, … |
| maximal quotient of | C3×#152071 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 5682);