Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2215 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C22 |
| direct product | C3×#11673 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C2×D8 |
| commutator | C2×C8 | C22×C3×C4 |
| Frattini, cosocle | C4×C8 | C23×C3 |
| socle | C22×C3 | C2×#102 |
| prime radical | C3×#466 | C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C2×C3×C42 ◁ C2×C3×C4×C8 ◁ G | |
332 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 3 generators on 43 points:
(1,2,3)(4,5,7,10,8,6,11,9)(12,13)(14,23)(15,20)(16,28)(17,32)(18,22)(19,25)(21,30)(24,39)(26,40)(27,38)(29,34)(31,42)(33,37)(35,43)(36,41);(4,6)(5,8)(7,9)(10,11)(12,14)(13,19)(15,24)(16,29)(17,33)(18,26)(20,34)(21,37)(22,36)(23,28)(25,30)(27,41)(31,43)(32,35)(38,39)(40,42);(5,6)(9,10)(12,15,18,17)(13,20,22,32)(14,24,26,33)(16,27,31,30)(19,34,36,35)(21,28,38,42)(23,39,40,37)(25,29,41,43)228 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 4661);