Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30428 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#30430 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4 ◁ G | |
878 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)(14,19)(15,27)(16,32)(17,22)(18,23)(20,29)(21,25)(24,38)(26,36)(28,34)(30,39)(31,40)(33,35)(37,41)(42,43);(4,6)(5,8)(7,11)(9,10)(12,14)(13,19)(15,28)(16,25)(17,33)(18,26)(20,37)(21,32)(22,35)(23,36)(24,30)(27,34)(29,41)(31,42)(38,39)(40,43);(12,15,18,31)(13,20,23,39)(14,24,26,41)(16,30,17,29)(19,34,36,43)(21,38,35,37)(22,40,32,27)(25,42,33,28);(5,9)(6,11)(12,14)(13,21)(15,24)(16,33)(17,25)(18,26)(19,32)(20,38)(22,36)(23,35)(27,34)(28,30)(29,42)(31,41)(37,39)(40,43)111 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 23418);