Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61643 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C28 |
| direct product | C3×#61670 |
| 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×#532 ◁ G | |
2102 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)(12,13,16,21)(14,23,24,18)(15,20,26,27)(17,19,25,22);(4,6)(5,8)(7,11)(9,10)(12,14)(13,18)(15,17)(16,24)(19,27)(20,22)(21,23)(25,26);(13,19)(14,24)(17,25)(18,20)(21,22)(23,27);(6,11)(8,10)(12,15)(13,20)(14,25)(16,26)(17,24)(18,19)(21,27)(22,23);(5,9)(8,10)(13,21)(14,24)(15,26)(19,22)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 54658);