Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25798 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25×C4 |
| direct product | C3×#25800 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | #2433 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C3×#3584 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C22×C3×C4 ◁ G | |
1402 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)(4,5,10,15)(6,16,20,41)(7,12,24,36)(8,25,28,55)(9,29,26,14)(11,30,33,19)(13,37,38,27)(17,42,21,39)(18,46,48,32)(22,51,53,67)(23,54,43,35)(31,45,58,50)(34,59,60,52)(40,56,62,47)(44,57,49,65)(61,64,63,66);(4,6,9,19,10,20,26,30)(5,11,14,16,15,33,29,41)(7,21,23,50,24,17,43,45)(8,18,27,47,28,48,37,56)(12,31,35,39,36,58,54,42)(13,32,25,40,38,46,55,62)(22,49,52,66,53,44,59,64)(34,57,51,61,60,65,67,63);(4,7)(5,12)(6,17,20,21)(8,22)(9,23)(10,24)(11,31,33,58)(13,34)(14,35)(15,36)(16,39,41,42)(18,44,48,49)(19,45,30,50)(25,51)(26,43)(27,52)(28,53)(29,54)(32,57,46,65)(37,59)(38,60)(40,61,62,63)(47,64,56,66)(55,67);(4,8,26,37,10,28,9,27)(5,13,29,55,15,38,14,25)(6,18,30,56,20,48,19,47)(7,22,43,59,24,53,23,52)(11,32,41,62,33,46,16,40)(12,34,54,67,36,60,35,51)(17,44,50,66,21,49,45,64)(31,57,42,63,58,65,39,61)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 18788);