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×#61672 |
| 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×#532 ◁ G | |
2018 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 5 generators on 43 points:
(1,2,3)(4,5)(6,10)(7,9)(8,11)(12,13,17,21)(14,18,23,33)(15,25,28,37)(16,29,31,20)(19,30,36,27)(22,38,26,43)(24,41,40,34)(32,42,35,39);(4,6)(5,8)(7,11)(9,10)(12,14)(13,18)(15,26)(16,24)(17,23)(19,35)(20,34)(21,33)(22,28)(25,43)(27,42)(29,41)(30,39)(31,40)(32,36)(37,38);(12,15,17,28)(13,19,21,36)(14,22,23,26)(16,30,31,27)(18,32,33,35)(20,37,29,25)(24,42,40,39)(34,43,41,38);(6,11)(8,10)(12,14)(13,18)(15,22)(16,24)(17,23)(19,32)(20,34)(21,33)(25,38)(26,28)(27,39)(29,41)(30,42)(31,40)(35,36)(37,43);(5,9)(8,10)(14,23)(18,33)(22,26)(24,40)(32,35)(34,41)(38,43)(39,42)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 54660);