Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33758 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#33763 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C3 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C23×C3×C4 ◁ C23×C3×C8 ◁ G | |
592 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 43 points:
(1,2,3)(4,5)(6,10)(7,9)(8,11)(12,13)(14,23)(15,20)(16,28)(17,32)(18,22)(19,26)(21,30)(24,39)(25,40)(27,38)(29,35)(31,42)(33,37)(34,41)(36,43);(4,6)(5,8)(7,11)(9,10)(14,24)(19,34)(23,39)(25,33)(26,41)(29,43)(35,36)(37,40);(6,11)(8,10)(12,14)(13,19)(15,25)(16,29)(17,33)(18,24)(20,35)(21,37)(22,34)(23,28)(26,30)(27,41)(31,43)(32,36)(38,40)(39,42);(5,9)(8,10)(12,15,18,17)(13,20,22,32)(14,25,24,33)(16,27,31,30)(19,35,34,36)(21,28,38,42)(23,40,39,37)(26,29,41,43)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26751);