Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61643 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C28 |
| direct product | C3×#61652 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C3 |
| prime radical | C3×#3487 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×#525 ◁ G | |
2030 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)(6,10)(7,9)(8,11)(12,13,18,23)(14,19,26,34)(15,27,29,37)(16,22,17,21)(20,31,35,28)(24,38,40,42)(25,36,30,33)(32,39,43,41);(4,6)(5,8)(7,11)(9,10)(12,14,18,26)(13,19,23,34)(15,24,29,40)(16,30,17,25)(20,32,35,43)(21,36,22,33)(27,38,37,42)(28,41,31,39);(12,15,18,29)(13,20,23,35)(14,24,26,40)(16,28,17,31)(19,32,34,43)(21,27,22,37)(25,39,30,41)(33,42,36,38);(6,11)(8,10)(12,16,18,17)(13,21,23,22)(14,25,26,30)(15,28,29,31)(19,33,34,36)(20,27,35,37)(24,39,40,41)(32,42,43,38);(5,9)(8,10)(14,26)(19,34)(24,40)(25,30)(32,43)(33,36)(38,42)(39,41)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 54640);