Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33522 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#33588 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C3 |
| prime radical | C2×C3×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×#117 ◁ C3×#405 ◁ G | |
680 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)(10,15,21,36)(11,24,27,58)(12,28,32,57)(13,33,30,18)(16,39,41,25)(17,42,44,31)(20,46,47,69)(22,52,50,68)(23,55,51,38)(26,59,34,43)(29,45,60,40)(35,61,62,48)(37,65,64,54)(49,70,56,66)(53,67,71,63);(4,5,6,7)(8,10)(9,15)(11,20)(12,22)(13,23)(14,21)(16,35)(17,37)(18,38)(19,36)(24,46)(25,48)(26,49)(27,47)(28,52)(29,53)(30,51)(31,54)(32,50)(33,55)(34,56)(39,61)(40,63)(41,62)(42,65)(43,66)(44,64)(45,67)(57,68)(58,69)(59,70)(60,71);(8,11,14,27)(9,16,19,41)(10,20,21,47)(12,29,32,60)(13,34,30,26)(15,35,36,62)(17,43,44,59)(18,45,33,40)(22,53,50,71)(23,56,51,49)(24,57,58,28)(25,42,39,31)(37,66,64,70)(38,67,55,63)(46,68,69,52)(48,65,61,54);(10,21)(15,36)(20,47)(22,50)(23,51)(35,62)(37,64)(38,55)(46,69)(48,61)(49,56)(52,68)(53,71)(54,65)(63,67)(66,70)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26576);