Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1902 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C8 · Schur multiplier (H2) = C24 |
| direct product | C3×#9090 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #386 |
| commutator | C2×C4 | C22×C3×C8 |
| Frattini, cosocle | C2×C42 | C23×C3 |
| prime radical | C22×C3×D4 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
512 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3)(4,5,7,10,8,6,9,11)(12,13)(14,17)(15,23,26,21)(16,22)(18,20)(19,25,27,24);(4,6,7,11,8,5,9,10)(12,14,17,13,18,22,16,20)(15,21,24,19,26,23,25,27);(12,15,18,26)(13,19,20,27)(14,21,22,23)(16,25,17,24)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 2078);