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×#61794 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C3 |
| prime radical | C3×#3547 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×#532 ◁ G | |
1674 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)(20,21,25,30)(22,26,31,39)(23,28,34,42)(24,27,36,29)(32,38,44,49)(33,40,45,37)(35,41,47,43)(46,50,51,48);(4,6)(5,9)(7,13)(8,14)(10,16)(11,12)(15,19)(17,18)(20,22)(21,26)(23,32)(24,33)(25,31)(27,40)(28,38)(29,37)(30,39)(34,44)(35,46)(36,45)(41,50)(42,49)(43,48)(47,51);(4,7,8,15)(5,10,11,17)(6,13,14,19)(9,16,12,18)(21,27)(22,31)(24,36)(26,37)(28,41)(29,30)(32,44)(35,47)(38,48)(39,40)(42,43)(49,50);(6,14)(9,12)(13,19)(16,18)(20,23)(21,28)(22,32)(24,35)(25,34)(26,38)(27,41)(29,43)(30,42)(31,44)(33,46)(36,47)(37,48)(39,49)(40,50)(45,51);(5,11)(9,12)(10,17)(16,18)(22,31)(23,34)(26,39)(28,42)(33,45)(35,47)(37,40)(41,43)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 54782);