Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32884 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C23 |
| direct product | C3×#33090 |
| 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 | |
770 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)(14,16)(15,17)(18,19)(20,21,24,28)(22,25,29,32)(23,26,31,27)(30,33,35,34);(4,6,8,7)(5,9,10,11)(12,14,15,18)(13,16,17,19)(20,22)(21,25)(23,30)(24,29)(26,33)(27,34)(28,32)(31,35);(4,7,8,6)(5,9,10,11)(13,17)(16,19)(22,29)(25,32)(30,35)(33,34);(21,26)(23,31)(25,33)(27,28)(30,35)(32,34)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26078);