Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19086 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#19095 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C3 |
| prime radical | C3×#3568 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
834 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 51 points:
(1,2,3)(6,13)(7,8)(10,17)(11,15)(14,16)(18,19)(20,21,26,31)(22,32,34,44)(23,36,38,45)(24,30,25,29)(27,39,42,35)(28,40,43,37)(33,41,47,51)(46,48,50,49);(4,5)(6,10)(7,11)(8,15)(9,12)(13,18)(14,19)(16,17)(20,22)(21,27)(23,33)(24,39)(25,35)(26,34)(28,41)(29,44)(30,32)(31,42)(36,50)(37,49)(38,47)(40,48)(43,51)(45,46);(4,6)(5,10)(7,13)(8,16)(9,14)(11,18)(12,19)(15,17)(20,23)(21,28)(22,33)(24,37)(25,40)(26,38)(27,41)(29,36)(30,45)(31,43)(32,46)(34,47)(35,48)(39,49)(42,51)(44,50);(4,7,9,8)(5,11,12,15)(6,13,14,16)(10,18,19,17)(22,34)(27,42)(32,44)(33,47)(35,39)(41,51)(46,50)(48,49)93 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 12083);