Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24264 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C22×C4 |
| direct product | C3×#27699 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C24×C3 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C3×C4×C22⋊C4 ◁ G | |
730 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2,3)(4,5)(6,9)(7,8)(10,11)(12,13)(14,16)(15,17)(18,19)(20,21,24,27,25,29,23,28)(22,26,31,34,32,35,30,33);(5,8)(9,11)(12,14,15,18)(13,16,17,19)(20,22,24,31,25,32,23,30)(21,26,27,34,29,35,28,33);(4,6,7,10)(5,9,8,11)(13,17)(16,19)(21,27)(22,30)(23,24)(28,29)(31,32)(33,34);(5,8)(9,11)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 20687);