Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1468 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C42 · Schur multiplier (H2) = C26 |
| direct product | C3×#10737 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C2×C22⋊C4 |
| commutator | C23 | C2×C3×C42 |
| Frattini, cosocle | C23×C4 | C23×C3 |
| socle | C22×C3 | C2×#95 |
| prime radical | C25×C3 | C2×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C24×C3×C4 ◁ G | |
1334 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)(10,13,16,20)(11,18)(12,14)(15,19)(17,23,22,21);(8,10,11,17,12,16,19,22)(9,13,15,21,14,20,18,23);(5,7)192 conjugacy classes — view the full character table.
| Fitting subgroup of | D3×#10737, #72249294, #72254086, … |
| maximal subgroup of | D3×#10737, C3×#374114, #72249294, … |
| maximal normal subgroup of | D3×#10737, C3×#374114, #72249294, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 3725);