Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2215 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#33600 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C3 |
| socle | C22×C3 | C22×D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C23×C3×C4 ◁ C23×C3×C8 ◁ G | |
710 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 39 points:
(1,2,3)(4,5)(6,7)(8,9)(10,19)(11,16)(12,26)(13,28)(14,18)(15,22)(17,20)(21,32)(23,38)(24,34)(25,31)(27,39)(29,36)(30,37)(33,35);(8,10,20,35,13,23,27,37,14,24,25,36,11,21,12,22)(9,15,26,32,16,29,31,34,18,30,39,38,28,33,17,19);(5,7);(8,11,14,13)(9,16,18,28)(10,21,24,23)(12,25,27,20)(15,29,30,33)(17,26,31,39)(19,32,34,38)(22,36,37,35)228 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26588);