Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30437 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#30440 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | #2668 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C3×#3020 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C22×C3×C4 ◁ G | |
850 conjugacy classes of subgroups.
| minimal degree | μ(G) = 67 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 67 points:
(1,2,3)(4,5)(6,18)(7,24)(8,14)(9,16)(10,15)(11,17)(12,21)(13,26)(19,37)(20,46)(22,36)(23,35)(25,54)(27,44)(28,59)(29,42)(30,41)(31,43)(32,39)(33,49)(34,50)(38,56)(40,51)(45,48)(47,60)(52,61)(53,57)(55,63)(58,64)(62,66)(65,67);(4,6,11,23)(5,12,17,36)(7,19,27,49)(8,28,31,50)(9,22,10,21)(13,37,39,33)(14,40,43,46)(15,35,16,18)(20,30,51,29)(24,45,44,61)(25,55,57,67)(26,52,32,48)(34,42,59,41)(38,63,64,65)(47,58,66,56)(53,60,54,62);(4,7)(5,13)(6,19)(8,25)(9,26)(10,32)(11,27)(12,33)(14,38)(15,24)(16,44)(17,39)(18,45)(20,47)(21,48)(22,52)(23,49)(28,55)(29,56)(30,58)(31,57)(34,60)(35,61)(36,37)(40,65)(41,54)(42,53)(43,64)(46,63)(50,67)(51,66)(59,62);(4,8,11,31)(5,14,17,43)(6,20,23,51)(7,25,27,57)(9,30,10,29)(12,34,36,59)(13,38,39,64)(15,42,16,41)(18,46,35,40)(19,47,49,66)(21,50,22,28)(24,53,44,54)(26,58,32,56)(33,60,37,62)(45,63,61,65)(48,67,52,55)102 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 23428);