Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19050 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#19057 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C3 |
| prime radical | C3×#3548 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
908 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2,3)(4,5,9,12)(6,13,15,18)(7,11,8,16)(10,17,19,14)(20,21,24,27)(22,28,29,35)(23,31,32,26)(25,33,34,30);(4,6,9,15)(5,10,12,19)(7,14,8,17)(11,18,16,13)(23,32)(26,31)(28,35)(30,33);(7,8)(11,16)(13,18)(14,17)(20,22)(21,25)(23,33)(24,29)(26,35)(27,34)(28,31)(30,32);(4,7,9,8)(5,11,12,16)(6,14,15,17)(10,18,19,13)102 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 12045);