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×#63035 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C26 |
| commutator, Frattini, cosocle | C22 | C26×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
7236 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)(4,5,9,8)(6,11,7,10)(14,21)(18,22)(23,26)(24,27);(4,6,9,7)(5,10,8,11)(12,13,17,20)(14,22,21,18)(15,16,19,25)(23,27,26,24);(4,6,9,7)(5,10,8,11)(12,14,17,21)(13,18,20,22)(15,23,19,26)(16,24,25,27);(4,6,9,7)(5,10,8,11);(4,7,9,6)(5,10,8,11)(12,15)(13,16)(14,23)(17,19)(18,24)(20,25)(21,26)(22,27);(4,8,9,5)(6,11,7,10)(12,16)(13,19)(14,24)(15,20)(17,25)(18,26)(21,27)(22,23)228 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 56023);