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×#19093 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C3 |
| prime radical | C3×#121 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
782 conjugacy classes of subgroups.
| minimal degree | μ(G) = 67 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 67 points:
(1,2,3)(4,5,10,15)(6,16,18,28)(7,20,22,29)(8,14,9,13)(11,23,26,19)(12,24,27,21)(17,25,31,35)(30,32,34,33)(36,37,41,44)(38,46,48,60)(39,43,51,58)(40,52,55,45)(42,53,57,49)(47,63,64,62)(50,61,54,59)(56,66,67,65);(4,6)(5,11)(7,17)(8,23)(9,19)(10,18)(12,25)(13,28)(14,16)(15,26)(20,34)(21,33)(22,31)(24,32)(27,35)(29,30)(36,38,41,48)(37,42,44,57)(39,47,51,64)(40,53,55,49)(43,56,58,67)(45,60,52,46)(50,66,54,65)(59,62,61,63);(4,7,10,22)(5,12,15,27)(6,17,18,31)(8,21,9,24)(11,25,26,35)(13,20,14,29)(16,30,28,34)(19,32,23,33)(36,39,41,51)(37,43,44,58)(38,47,48,64)(40,54,55,50)(42,56,57,67)(45,61,52,59)(46,62,60,63)(49,66,53,65);(6,18)(11,26)(16,28)(17,31)(19,23)(25,35)(30,34)(32,33)(37,44)(38,48)(43,58)(45,52)(47,64)(49,53)(59,61)(65,66)93 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 12081);