Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24274 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C2×C42 |
| direct product | C3×#27704 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C2×C3×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C3×C4×C22⋊C4 ◁ G | |
784 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(4,5)(6,9)(7,8)(10,11)(12,13)(14,16)(15,17)(18,19)(20,21,25,26,24,22,23,27);(5,8)(9,11)(12,14,15,18)(13,16,17,19)(20,22,23,26,24,21,25,27);(4,6,7,10)(5,9,8,11)(13,17)(16,19)(20,22,24,21)(23,27,25,26);(5,8)(9,11)(20,23,24,25)(21,26,22,27)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 20692);