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×#61806 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C3 |
| prime radical | C3×#532 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×#532 ◁ G | |
1668 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 384 |
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)(20,21,24,29)(22,31,33,26)(23,28,35,42)(25,27,37,30)(32,44,45,39)(34,40,47,38)(36,41,48,43)(46,50,51,49);(4,6)(5,9)(7,13)(8,14)(10,16)(11,12)(15,19)(17,18)(20,22,24,33)(21,26,29,31)(23,32,35,45)(25,34,37,47)(27,38,30,40)(28,39,42,44)(36,46,48,51)(41,49,43,50);(4,7,8,15)(5,10,11,17)(6,13,14,19)(9,16,12,18)(21,27)(25,37)(26,38)(28,41)(29,30)(31,40)(34,47)(36,48)(39,49)(42,43)(44,50)(46,51);(6,14)(9,12)(13,19)(16,18)(20,23)(21,28)(22,32)(24,35)(25,36)(26,39)(27,41)(29,42)(30,43)(31,44)(33,45)(34,46)(37,48)(38,49)(40,50)(47,51);(5,11)(9,12)(10,17)(16,18)(21,29)(22,33)(23,35)(27,30)(34,47)(36,48)(39,44)(49,50)147 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 54794);