Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25140 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#25491 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C23×C3 | C2×#41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
632 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(4,5,8,11)(6,10,7,9)(12,13,18,20,17,15,16,21)(14,22,25,27,24,19,23,26);(4,6,8,7)(5,9,11,10)(12,14,18,25,17,24,16,23)(13,19,20,26,15,22,21,27);(12,15,17,13)(14,22,24,19)(16,21,18,20)(23,27,25,26);(4,7,8,6)(5,10,11,9)(12,16,17,18)(13,20,15,21)(14,23,24,25)(19,26,22,27)129 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 18479);