Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32944 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C23×C4 |
| direct product | C3×#33065 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C2×C3×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
746 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)(12,13,19,25)(14,21,28,35)(15,20,27,36)(16,30,33,38)(17,24,18,23)(22,32,39,31)(26,40,42,34)(29,37,41,43);(5,8)(9,11)(12,14,17,15,19,28,18,27)(13,20,23,21,25,36,24,35)(16,26,31,29,33,42,32,41)(22,34,30,37,39,40,38,43);(4,6,7,10)(5,9,8,11)(12,15,17,28,19,27,18,14)(13,21,23,36,25,35,24,20)(16,29,31,42,33,41,32,26)(22,37,30,40,39,43,38,34);(12,16)(13,22)(14,26)(15,29)(17,31)(18,32)(19,33)(20,34)(21,37)(23,30)(24,38)(25,39)(27,41)(28,42)(35,43)(36,40)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26053);