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×#61801 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C3 |
| prime radical | C3×#3551 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×#532 ◁ G | |
1692 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 51 points:
(1,2,3)(4,5)(6,12)(7,10)(8,11)(9,14)(13,18)(15,17)(16,19)(20,21)(22,26)(23,28)(24,27)(25,30)(29,37)(31,39)(32,40)(33,38)(34,43)(35,44)(36,42)(41,47)(45,49)(46,50)(48,51);(4,6)(5,9)(7,13)(8,14)(10,16)(11,12)(15,19)(17,18)(20,22)(21,26)(23,34)(24,33)(25,32)(27,38)(28,43)(29,41)(30,40)(31,35)(36,48)(37,47)(39,44)(42,51)(45,50)(46,49);(4,7,8,15)(5,10,11,17)(6,13,14,19)(9,16,12,18)(21,27)(24,37)(26,38)(28,42)(29,30)(33,47)(36,49)(39,50)(40,41)(43,51)(44,45)(46,48);(6,14)(9,12)(13,19)(16,18)(20,23)(21,28)(22,31)(24,36)(25,35)(26,39)(27,42)(29,45)(30,44)(32,34)(33,46)(37,49)(38,50)(40,43)(41,51)(47,48);(5,11)(9,12)(10,17)(16,18)(22,32)(23,35)(26,40)(28,44)(33,47)(36,49)(38,41)(42,45)147 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 54789);