Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3198 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C24 |
| direct product | C3×#17993 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C22×D4 |
| commutator | C2×C4 | C23×C3×C4 |
| Frattini, cosocle | C42 | C24×C3 |
| socle | C22×C3 | C2×C4×D4 |
| prime radical | C3×#466 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C3×#431 ◁ G | |
576 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)(4,5,10,15)(6,16,7,12)(8,14,9,13)(11,18,19,17)(20,21,24,28,25,27,33,34)(22,26,30,23,29,31,35,32);(4,6,10,7)(5,11,15,19)(8,17,9,18)(12,13,16,14)(20,22)(21,26)(23,28)(24,30)(25,29)(27,31)(32,34)(33,35);(4,7,10,6)(5,12,15,16)(8,17,9,18)(11,13,19,14)(20,23,24,31,25,32,33,26)(21,22,28,30,27,29,34,35);(4,8,10,9)(5,13,15,14)(6,17,7,18)(11,16,19,12)(21,27)(22,29)(28,34)(30,35)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 10981);