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×#63018 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C26 |
| commutator, Frattini, cosocle | C22 | C26×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
7282 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)(14,21)(15,23)(16,18)(17,20)(19,25)(22,26)(24,27);(4,5)(6,9)(7,10)(8,11)(12,14,17,16)(13,18,20,21)(15,22,25,24)(19,26,23,27);(4,6)(5,8)(7,11)(9,10)(12,14,17,16)(13,18,20,21)(15,22,25,24)(19,26,23,27);(12,15)(13,19)(14,22)(16,24)(17,25)(18,26)(20,23)(21,27);(4,6)(5,9)(7,11)(8,10)(12,16,17,14)(13,18,20,21)(15,24,25,22)(19,26,23,27);(4,5)(6,8)(7,10)(9,11)(12,13)(14,18)(15,19)(16,21)(17,20)(22,26)(23,25)(24,27)228 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 56006);