Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61609 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C28 |
| direct product | C3×#61626 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C3 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×#525 ◁ G | |
2046 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)(14,19)(15,27)(16,21)(17,30)(18,22)(20,28)(23,38)(24,33)(25,34)(26,35)(29,36)(31,37)(32,39)(40,43)(41,42);(4,6)(5,8)(7,11)(9,10)(12,14,18,26)(13,19,22,35)(15,23,29,41)(16,24,17,25)(20,32,37,43)(21,33,30,34)(27,38,36,42)(28,39,31,40);(12,15)(13,20)(14,23)(16,28)(17,31)(18,29)(19,32)(21,36)(22,37)(24,39)(25,40)(26,41)(27,30)(33,42)(34,38)(35,43);(6,11)(8,10)(12,16,18,17)(13,21,22,30)(14,24,26,25)(15,28,29,31)(19,33,35,34)(20,36,37,27)(23,39,41,40)(32,42,43,38);(5,9)(8,10)(12,16,18,17)(13,21,22,30)(14,25,26,24)(15,28,29,31)(19,34,35,33)(20,36,37,27)(23,40,41,39)(32,38,43,42)147 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 54614);