Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33522 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26 |
| direct product | C3×#33525 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C3 |
| prime radical | C2×C3×#578 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×#41 ◁ C2×C3×#131 ◁ G | |
1086 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)(4,5)(6,7)(8,9,14,19)(10,20,24,48)(11,16,25,39)(12,28,32,46)(13,33,30,18)(15,35,38,22)(17,42,44,31)(21,47,49,69)(23,52,34,43)(26,55,53,68)(27,58,54,41)(29,45,60,37)(36,61,62,50)(40,65,64,57)(51,70,59,66)(56,67,71,63);(8,10,14,24)(9,15,19,38)(11,21,25,49)(12,29,32,60)(13,34,30,23)(16,36,39,62)(17,43,44,52)(18,45,33,37)(20,46,48,28)(22,42,35,31)(26,56,53,71)(27,59,54,51)(40,66,64,70)(41,67,58,63)(47,68,69,55)(50,65,61,57);(5,7)(8,11)(9,16)(10,21)(12,26)(13,27)(14,25)(15,36)(17,40)(18,41)(19,39)(20,47)(22,50)(23,51)(24,49)(28,55)(29,56)(30,54)(31,57)(32,53)(33,58)(34,59)(35,61)(37,63)(38,62)(42,65)(43,66)(44,64)(45,67)(46,68)(48,69)(52,70)(60,71);(11,25)(16,39)(21,49)(26,53)(27,54)(36,62)(40,64)(41,58)(47,69)(50,61)(51,59)(55,68)(56,71)(57,65)(63,67)(66,70)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26513);