Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #7855 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C42 · Schur multiplier (H2) = C25×C4 |
| direct product | C3×#10717 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C2×C42 |
| commutator | C2×C4 | C2×C3×C42 |
| Frattini, cosocle | C23×C4 | C23×C3 |
| socle | C22×C3 | C2×#95 |
| prime radical | C25×C3 | D4 |
| LCS | G ▷ C2×C4 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ G | |
1374 conjugacy classes of subgroups.
| 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)(10,17,19,23)(11,15)(12,21)(13,16)(14,18,22,20);(8,10,12,20,13,19,11,18)(9,14,15,17,16,22,21,23);(5,7)192 conjugacy classes — view the full character table.
| Fitting subgroup of | D3×#10717, #72249323, #72253916, … |
| maximal subgroup of | D3×#10717, C3×#364424, #72249323, … |
| maximal normal subgroup of | D3×#10717, C3×#364424, #72249323, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 3705);