Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2192 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26 |
| direct product | C2×C3×#2195 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #447 |
| commutator, Frattini, cosocle | C2×C8 | C24×C3 |
| prime radical | C2×C3×#578 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C2×C3×#131 ◁ G | |
1078 conjugacy classes of subgroups.
| minimal degree | μ(G) = 69 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 69 points:
(1,2)(67,68,69);(3,4,10,16)(5,17,21,45)(6,22,25,42)(7,26,30,40)(8,15,9,14)(11,33,36,19)(12,31,38,24)(13,28,41,29)(18,46,50,63)(20,51,32,39)(23,54,57,37)(27,43,58,35)(34,55,62,66)(44,49,65,48)(47,52,60,61)(53,59,64,56);(3,5)(4,11)(6,18)(7,27)(8,20)(9,32)(10,21)(12,34)(13,39)(14,35)(15,43)(16,36)(17,26)(19,29)(22,52)(23,55)(24,48)(25,50)(28,33)(30,58)(31,49)(37,63)(38,62)(40,45)(41,51)(42,61)(44,53)(46,54)(47,56)(57,66)(59,60)(64,65);(3,6,10,25)(4,12,16,38)(5,18,21,50)(7,23,30,57)(8,31,9,24)(11,34,36,62)(13,37,41,54)(14,42,15,22)(17,44,45,65)(19,47,33,60)(20,49,32,48)(26,53,40,64)(27,55,58,66)(28,59,29,56)(35,61,43,52)(39,63,51,46)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26525);