Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #63011 |
| automorphism | ? |
| homology | Abelianization (H1) = C26×C3 · Schur multiplier (H2) = C213 |
| direct product | C3×#63020 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C26 |
| commutator, Frattini, cosocle | C22 | C26×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
7272 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 6 generators on 27 points:
(1,2,3)(4,5)(6,9)(7,10)(8,11)(12,13,19,17)(14,20,16,15)(18,21,26,25)(22,27,24,23);(4,5)(6,9)(7,10)(8,11)(12,14,19,16)(13,15,17,20)(18,22,26,24)(21,23,25,27);(4,6)(5,8)(7,11)(9,10)(12,15)(13,16)(14,17)(18,23)(19,20)(21,24)(22,25)(26,27);(12,16,19,14)(13,20,17,15)(18,24,26,22)(21,27,25,23);(4,6)(5,9)(7,11)(8,10)(12,17,19,13)(14,20,16,15)(18,21,26,25)(22,23,24,27);(4,5)(6,8)(7,10)(9,11)(12,18)(13,21)(14,22)(15,23)(16,24)(17,25)(19,26)(20,27)228 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 56008);