Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60454 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C27×C4 |
| direct product | C3×#60620 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C3 |
| prime radical | C22×C3×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×#117 ◁ G | |
2014 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 43 points:
(1,2,3)(4,5)(6,9)(7,8)(10,11)(12,13,18,23)(14,24,26,35)(15,29,30,37)(16,22,17,21)(19,28,34,27)(20,31,36,32)(25,33,39,43)(38,40,42,41);(5,8)(9,11)(12,14,17,28,18,26,16,27)(13,19,22,35,23,34,21,24)(15,25,32,41,30,39,31,40)(20,33,29,42,36,43,37,38);(4,6,7,10)(5,9,8,11)(12,15,16,31,18,30,17,32)(13,20,21,37,23,36,22,29)(14,25,27,40,26,39,28,41)(19,33,24,38,34,43,35,42);(15,30)(20,36)(25,39)(29,37)(31,32)(33,43)(38,42)(40,41);(14,26)(19,34)(24,35)(25,39)(27,28)(33,43)(38,42)(40,41)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 53608);