Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2215 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#33726 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C3 |
| socle | C22×C3 | C22×D8 |
| prime radical | C23×C3 | C2×D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C2×C3×C42 ◁ C2×C3×C4×C8 ◁ G | |
490 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 43 points:
(1,2,3)(4,5,7,9)(6,10,11,8)(12,13)(14,23)(15,20)(16,30)(17,32)(18,22)(19,26)(21,24)(25,36)(27,42)(28,38)(29,35)(31,43)(33,40)(34,41)(37,39);(4,6,7,11)(5,8,9,10);(12,14,24,39,17,27,31,41,18,28,29,40,15,25,16,26)(13,19,30,36,20,33,35,38,22,34,43,42,32,37,21,23);(12,15,18,17)(13,20,22,32)(14,25,28,27)(16,29,31,24)(19,33,34,37)(21,30,35,43)(23,36,38,42)(26,40,41,39)228 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26714);