Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C8 · Schur multiplier (H2) = C25 |
| direct product | C3×#33442 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C8 | C22×D4 |
| commutator | C4 | C23×C3×C8 |
| Frattini, cosocle | C2×C8 | C24×C3 |
| socle | C2×C3 | C2×C8×D4 |
| prime radical | C3×#575 | C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C8 ◁ C2×C3×M32 ◁ G | |
642 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)(4,5,9,15,10,14,29,39,11,16,30,40,8,26,28,52)(6,17,20,42,19,12,31,33,21,34,46,58,27,35,53,59)(7,13,24,38,25,37,51,61,22,36,47,60,23,48,50,64)(18,41,45,62,44,32,54,56,43,57,63,67,49,55,65,66);(4,6)(5,12)(7,18)(8,27)(9,20)(10,19)(11,21)(13,32)(14,34)(15,33)(16,35)(17,26)(22,43)(23,49)(24,45)(25,44)(28,53)(29,31)(30,46)(36,55)(37,57)(38,56)(39,58)(40,59)(41,48)(42,52)(47,63)(50,65)(51,54)(60,66)(61,67)(62,64);(4,7,9,24,10,25,29,51,11,22,30,47,8,23,28,50)(5,13,15,38,14,37,39,61,16,36,40,60,26,48,52,64)(6,18,20,45,19,44,31,54,21,43,46,63,27,49,53,65)(12,32,33,56,34,57,58,67,35,55,59,66,17,41,42,62);(7,22)(13,36)(18,43)(23,25)(24,47)(32,55)(37,48)(38,60)(41,57)(44,49)(45,63)(50,51)(54,65)(56,66)(61,64)(62,67)264 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26430);