Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1504 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C26 |
| direct product | C3×#17572 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C2×C22⋊C4 |
| commutator | C2×C4 | C23×C3×C4 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| socle | C22×C3 | C22×C22⋊C4 |
| prime radical | C2×C3×#525 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C24×C3×C4 ◁ G | |
1470 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(4,5)(6,10,11,8)(7,9)(12,13)(14,22,24,27)(15,20)(16,21)(17,19)(18,23,26,25);(4,6)(5,8)(7,11)(9,10)(12,14)(13,18)(15,23)(16,25)(17,24)(19,26)(20,22)(21,27);(5,9)(8,10)(13,19)(18,26)(20,21)(22,27);(12,15,17,16)(13,20,19,21)(14,23,24,25)(18,22,26,27)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 10560);