Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25140 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#25494 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C23×C3 | C2×#41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
616 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)(4,5,8,11)(6,10,7,9)(12,13,18,32,16,21,17,22)(14,23,33,35,25,19,26,39)(15,27,31,20,29,38,30,37)(24,40,42,36,28,43,41,34);(4,6,8,7)(5,9,11,10)(12,14,18,33,16,25,17,26)(13,19,32,39,21,23,22,35)(15,28,31,41,29,24,30,42)(20,36,38,43,37,34,27,40);(12,15,16,29)(13,20,21,37)(14,24,25,28)(17,31,18,30)(19,34,23,36)(22,38,32,27)(26,42,33,41)(35,40,39,43);(4,7,8,6)(5,10,11,9)(13,21)(19,23)(20,37)(22,32)(27,38)(34,36)(35,39)(40,43)129 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 18482);