Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23924 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C23 |
| direct product | C3×#27620 |
| 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 ◁ C2×C3×#117 ◁ G | |
670 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)(6,11)(7,9)(8,10)(12,13,15,17)(14,16,18,19)(20,21)(22,27)(23,32)(24,28)(25,33)(26,29)(30,34)(31,35);(4,6,8,7)(5,9,10,11)(12,14)(13,16)(15,18)(17,19)(20,22)(21,27)(23,24)(25,30)(26,31)(28,32)(29,35)(33,34);(4,7,8,6)(5,9,10,11)(14,18)(16,19)(20,23)(21,27)(22,24)(25,31)(26,30)(28,34)(29,35)(32,33);(5,10)(9,11)(20,24,26,25)(21,28,29,33)(22,30,31,23)(27,34,35,32)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 20608);