Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33607 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24×C4 |
| direct product | C3×#33721 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C3 |
| prime radical | C2×C3×#508 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×#117 ◁ C3×#405 ◁ G | |
794 conjugacy classes of subgroups.
| minimal degree | μ(G) = 71 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 71 points:
(1,2,3)(5,7)(8,9,14,19)(10,20,25,45)(11,16,26,37)(12,29,31,57)(13,32,30,18)(15,33,36,23)(17,40,41,21)(22,42,48,64)(24,52,51,44)(27,55,53,70)(28,56,54,39)(34,58,60,49)(35,46,43,59)(38,63,62,47)(50,69,68,66)(61,67,65,71);(8,10,21,46,13,24,31,33,14,25,40,59,30,51,12,23)(9,15,29,44,18,35,41,45,19,36,57,52,32,43,17,20)(11,22,47,67,28,50,53,58,26,48,63,71,54,68,27,49)(16,34,55,66,39,61,62,64,37,60,70,69,56,65,38,42);(4,5,6,7)(8,11)(9,16)(10,22)(12,27)(13,28)(14,26)(15,34)(17,38)(18,39)(19,37)(20,42)(21,47)(23,49)(24,50)(25,48)(29,55)(30,54)(31,53)(32,56)(33,58)(35,61)(36,60)(40,63)(41,62)(43,65)(44,66)(45,64)(46,67)(51,68)(52,69)(57,70)(59,71);(11,26)(16,37)(22,48)(27,53)(28,54)(34,60)(38,62)(39,56)(42,64)(47,63)(49,58)(50,68)(55,70)(61,65)(66,69)(67,71)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26709);