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×#19089 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C3 |
| prime radical | C3×#3559 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
850 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 51 points:
(1,2,3)(4,5)(6,12)(7,15)(8,10)(9,14)(11,16)(13,19)(17,18)(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);(7,16)(11,15)(12,19)(14,17)(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)(5,9)(7,17)(8,13)(10,18)(11,19)(12,15)(14,16)(20,23,26,38)(21,28,31,43)(22,33,34,47)(24,37,25,40)(27,41,42,51)(29,36,30,45)(32,46,44,50)(35,48,39,49);(5,10)(9,18)(11,15)(12,19)(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, 12077);