Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #62892 |
| automorphism | ? |
| homology | Abelianization (H1) = C26×C3 · Schur multiplier (H2) = C213 |
| direct product | C3×#62955 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C26 |
| commutator, Frattini, cosocle | C22 | C26×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
7310 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)(6,7)(8,9)(10,11)(12,13)(14,21)(15,18)(16,19)(17,23)(20,25)(22,26)(24,27);(5,9)(6,10)(12,14,18,16)(13,19,15,21)(17,22,25,24)(20,26,23,27);(4,6,8,10)(5,7,9,11)(12,13,18,15)(14,21,16,19)(17,20,25,23)(22,27,24,26);(4,7)(5,10)(6,9)(8,11)(12,15,18,13)(14,21,16,19)(17,23,25,20)(22,27,24,26);(12,16,18,14)(13,19,15,21)(17,24,25,22)(20,26,23,27);(12,17)(13,20)(14,22)(15,23)(16,24)(18,25)(19,26)(21,27)219 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 55943);