Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C28 |
| direct product | C3×#60676 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C42 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C3 |
| socle | C22×C3 | C23×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C42 ◁ C22×C3×C42 ◁ G | |
1998 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 27 points:
(1,2,3)(4,5)(6,9)(7,8)(10,11)(12,13)(14,21)(15,19)(16,25)(17,20)(18,24)(22,27)(23,26);(5,8)(9,11)(14,22)(18,26)(21,27)(23,24);(12,14,16,24,17,22,15,23)(13,18,19,21,20,26,25,27);(4,6,7,10)(5,9,8,11);(12,15,17,16)(13,19,20,25)(14,23,22,24)(18,21,26,27)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 53664);