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×#63024 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C26 |
| commutator, Frattini, cosocle | C22 | C26×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
7264 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)(6,10)(7,8)(9,11)(12,13,17,21)(14,16,20,18)(15,23,25,19)(22,26,27,24);(5,8)(9,10)(12,14)(13,18)(15,22)(16,21)(17,20)(19,26)(23,24)(25,27);(4,5)(6,9)(7,8)(10,11)(12,14)(13,18)(15,22)(16,21)(17,20)(19,26)(23,24)(25,27);(4,6,7,11)(5,9,8,10)(12,15,17,25)(13,19,21,23)(14,22,20,27)(16,24,18,26);(12,14)(13,16)(15,22)(17,20)(18,21)(19,24)(23,26)(25,27);(12,16)(13,20)(14,21)(15,24)(17,18)(19,27)(22,23)(25,26)228 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 56012);