Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C27 |
| direct product | C2×C3×#2740 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C3 | C24 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C23×C3×D4 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24×C3 ◁ G | |
2330 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 5 generators on 29 points:
(1,2)(27,28,29);(3,4)(5,7)(6,8)(9,10)(11,12,14,16)(13,17,18,15)(20,23)(24,25);(4,5)(6,10)(11,12,14,16)(13,15,18,17)(19,20,22,23)(21,25,26,24);(3,5,7,4)(6,9,10,8)(11,12)(13,15)(14,16)(17,18)(19,21,22,26)(20,24,23,25);(3,6)(4,8)(5,9)(7,10)(11,13)(12,15)(14,18)(16,17)(19,20)(21,24)(22,23)(25,26)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 45163);