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×#2742 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C3 | C24 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C25×C3 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24×C3 ◁ G | |
2222 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);(3,4,7,6)(5,9,10,8)(11,12)(13,17)(14,16)(15,18)(19,20,23,25)(21,22,26,24);(4,6)(8,9)(11,12,14,16)(13,17,18,15)(19,21,23,26)(20,24,25,22);(3,5)(4,8)(6,9)(7,10)(11,12)(13,15)(14,16)(17,18);(3,6,7,4)(5,9,10,8)(11,13,14,18)(12,15,16,17)(19,22,23,24)(20,21,25,26)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 45165);