Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1619 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C22 |
| direct product | C3×#8901 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | #120 |
| commutator | C42 | C22×C3×C4 |
| Frattini, cosocle | C2×C42 | C23×C3 |
| socle | C22×C3 | #385 |
| prime radical | C2×C3×#41 | Q8 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C2×C3×C42 ◁ G | |
372 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2,3)(4,5,8,11,9,12,7,16)(6,13,14,10,15,18,17,19)(20,21,24,27)(22,28,29,35)(23,31,32,26)(25,33,34,30);(4,6,8,14,9,15,7,17)(5,10,11,18,12,19,16,13)(23,32)(26,31)(28,35)(30,33);(20,22,24,29)(21,25,27,34)(23,33,32,30)(26,35,31,28)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1889);