Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18252 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#18262 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C3×#3534 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G | |
1012 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2,3)(4,5,9,7)(6,14,18,30)(8,21,23,33)(10,12,25,19)(11,26,15,16)(13,28,20,24)(17,29,32,34)(22,27,35,31);(4,6)(5,11)(7,15)(8,22)(9,18)(10,17)(12,27)(13,29)(14,21)(16,24)(19,31)(20,34)(23,35)(25,32)(26,28)(30,33);(4,7)(5,9)(6,15)(8,13)(10,19)(11,18)(12,25)(14,16)(17,31)(20,23)(21,24)(22,29)(26,30)(27,32)(28,33)(34,35);(5,12)(7,19)(11,27)(13,21)(14,29)(15,31)(20,33)(30,34)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 11250);