Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32898 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C29 |
| direct product | C3×#32899 |
| 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 | |
3866 conjugacy classes of subgroups; only the 366 normal subgroups are shown.
This group has 3866 conjugacy classes of subgroups — too many to draw. Only its 366 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 — 4 generators on 23 points:
(1,2,3)(4,5)(6,7)(13,19)(20,23);(8,9)(10,13)(11,14)(12,15)(16,20)(17,19)(18,21)(22,23);(8,10)(9,13)(11,16)(12,17)(14,20)(15,19)(18,22)(21,23);(5,7)(8,11,12,18)(9,14,15,21)(10,16,17,22)(13,20,19,23)147 conjugacy classes — view the full character table.
| Fitting subgroup of | #72271216, D3×#32899 |
| maximal subgroup of | #72271216, D3×#32899, C3×#6628896 |
| maximal normal subgroup of | #72271216, D3×#32899, C3×#6628896 |
| maximal quotient of | C3×#511853 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 25887);