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 |
| direct product | C3×#33741 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C3 |
| prime radical | C23×C3 | C2×D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×#117 ◁ C3×#405 ◁ G | |
484 conjugacy classes of subgroups.
| minimal degree | μ(G) = 75 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 75 points:
(1,2,3)(4,5)(6,9)(7,8)(10,11)(12,13,18,23)(14,19,25,38)(15,28,32,57)(16,33,35,61)(17,36,34,22)(20,41,43,30)(21,44,45,29)(24,46,48,68)(26,53,51,74)(27,54,52,40)(31,60,59,56)(37,62,63,49)(39,66,65,47)(42,58,55,67)(50,73,72,70)(64,71,69,75);(5,8)(9,11)(12,14,18,25)(13,19,23,38)(15,24,32,48)(16,26,35,51)(17,27,34,52)(20,37,43,63)(21,39,45,65)(22,40,36,54)(28,46,57,68)(29,47,44,66)(30,49,41,62)(31,50,59,72)(33,53,61,74)(42,64,55,69)(56,70,60,73)(58,71,67,75);(12,15,29,58,17,31,35,41,18,32,44,67,34,59,16,30)(13,20,33,56,22,42,45,57,23,43,61,60,36,55,21,28)(14,24,47,71,27,50,51,62,25,48,66,75,52,72,26,49)(19,37,53,70,40,64,65,68,38,63,74,73,54,69,39,46);(4,6,7,10)(5,9,8,11)(14,25)(19,38)(24,48)(26,51)(27,52)(37,63)(39,65)(40,54)(46,68)(47,66)(49,62)(50,72)(53,74)(64,69)(70,73)(71,75)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26729);