Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #3415 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C25 |
| direct product | C3×#33465 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | C22×D8 |
| commutator | C8 | C23×C3×C4 |
| Frattini, cosocle | C2×C8 | C24×C3 |
| socle | C2×C3 | C2×C4×D8 |
| prime radical | C2×C3×#508 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C2×C3×M16 ◁ C3×#405 ◁ 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)(4,5,9,16,11,14,10,31)(6,17,20,44,21,43,32,41)(7,13,24,38,22,39,25,54)(8,26,28,40,30,56,29,15)(12,27,35,48,34,57,36,19)(18,42,47,63,45,64,55,61)(23,50,52,62,49,66,53,37)(33,51,58,65,60,67,59,46);(4,6)(5,12)(7,18)(8,27)(9,20)(10,32)(11,21)(13,33)(14,34)(15,41)(16,35)(17,26)(19,29)(22,45)(23,51)(24,47)(25,55)(28,48)(30,57)(31,36)(37,61)(38,58)(39,60)(40,44)(42,50)(43,56)(46,53)(49,67)(52,65)(54,59)(62,63)(64,66);(4,7,9,24,11,22,10,25)(5,13,16,38,14,39,31,54)(6,18,20,47,21,45,32,55)(8,23,28,52,30,49,29,53)(12,33,35,58,34,60,36,59)(15,37,26,50,40,62,56,66)(17,42,44,63,43,64,41,61)(19,46,27,51,48,65,57,67);(5,14)(7,22)(12,34)(15,40)(16,31)(17,43)(18,45)(23,49)(24,25)(26,56)(35,36)(41,44)(46,65)(47,55)(51,67)(52,53)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26453);