Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24412 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#25827 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C23×C3 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
910 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(4,5,8,11)(6,10,7,9)(12,13)(14,19)(15,22)(16,20)(17,26)(18,21)(23,24)(25,27);(4,6,8,7)(5,9,11,10)(12,14)(13,19)(15,20)(16,22)(17,27)(18,23)(21,24)(25,26);(12,15,17,19,18,25,16,24)(13,14,20,22,21,23,26,27);(4,7,8,6)(5,10,11,9)(12,16,18,17)(13,20,21,26)(14,22,23,27)(15,24,25,19)129 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 18815);