Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10012 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C3×#10033 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | #127 |
| commutator, Frattini, cosocle | C2×C42 | C23×C3 |
| socle | C22×C3 | #392 |
| prime radical | C3×#3100 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C2×C3×C42 ◁ G | |
550 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)(4,5,8,6)(7,10,9,11)(12,13)(14,20)(15,23)(16,19)(17,22)(18,25)(21,27)(24,26);(5,10)(6,11)(7,9)(17,26)(20,27);(4,6)(5,8)(7,11)(9,10)(12,14)(13,17)(15,24)(16,21)(18,27)(19,26)(20,23)(22,25)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 3021);