Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33607 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#33740 |
| 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 ◁ C3×#117 ◁ C3×#405 ◁ G | |
532 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)(6,9)(7,8)(10,11)(14,22)(15,23)(16,26)(18,28)(20,29)(21,33)(24,38)(25,36)(27,37)(30,39)(31,43)(32,42)(34,35)(40,41);(5,8)(9,11)(12,13,17,19)(14,18,25,30)(15,20,27,32)(16,21,26,33)(22,28,36,39)(23,29,37,42)(24,31,34,40)(35,41,38,43);(12,14,23,35,16,24,27,36,17,25,37,38,26,34,15,22)(13,18,29,41,21,31,32,39,19,30,42,43,33,40,20,28);(4,6,7,10)(5,9,8,11)(13,19)(18,30)(20,32)(21,33)(28,39)(29,42)(31,40)(41,43)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26728);