Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61609 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C28 |
| direct product | C3×#61625 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C3 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×#525 ◁ G | |
2118 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)(6,10)(7,9)(8,11)(12,13)(14,23)(15,20)(16,22)(17,21)(18,25)(19,24)(26,27);(4,6)(5,8)(7,11)(9,10)(12,14)(13,18)(15,26)(16,25)(17,24)(19,20)(21,27)(22,23);(12,14)(13,19)(15,24)(16,25)(17,26)(18,20)(21,23)(22,27);(6,11)(8,10)(12,15,16,17)(13,20,22,21)(14,24,25,26)(18,27,23,19);(5,9)(8,10)(12,15,16,17)(13,21,22,20)(14,24,25,26)(18,19,23,27)147 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 54613);