Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18618 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#18680 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C3 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
1000 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(5,9)(6,7)(10,11)(12,13)(14,22)(15,21)(16,19)(17,20)(18,24)(23,26)(25,27);(4,5,8,11)(6,9,7,10)(12,14,17,23)(13,18,20,27)(15,25,16,24)(19,26,21,22);(4,6,8,7)(5,9,11,10)(13,19)(15,16)(18,26)(20,21)(22,27)(24,25);(4,6,8,7)(5,10,11,9)(13,20)(14,23)(19,21)(24,25)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 11668);