Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24390 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C22×C4 |
| direct product | C3×#29511 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C22×C3×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C3×#376 ◁ G | |
724 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)(6,10)(7,9)(8,11)(20,21,23,26)(22,28,30,35)(24,32,29,27)(25,33,34,31);(4,6,7,11)(5,8,9,10)(12,13)(14,17)(15,16)(18,19)(20,21)(22,25)(23,26)(24,27)(28,31)(29,32)(30,34)(33,35);(13,16)(17,19)(20,22,29,33,23,30,24,31)(21,25,32,35,26,34,27,28);(5,9)(8,10)(12,14,15,18)(13,17,16,19)(21,26)(25,34)(27,32)(28,35)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 22499);