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×#61673 |
| 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 | |
2160 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)(13,19)(14,21)(16,18)(20,23)(22,27)(24,26);(4,6)(5,8)(7,11)(9,10)(12,13)(14,22)(15,18)(16,25)(17,19)(20,24)(21,23)(26,27);(12,14)(13,20)(15,23)(16,21)(17,24)(18,26)(19,22)(25,27);(6,11)(8,10)(12,15)(13,16)(14,23)(17,25)(18,19)(20,21)(22,26)(24,27);(5,9)(8,10)(12,16,17,18)(13,15,19,25)(14,21,24,26)(20,23,22,27)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 54661);