Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18224 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#18241 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C3×#3558 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
992 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2,3)(4,5,8,12)(6,10,13,17)(7,9,15,11)(14,16,19,18)(25,31)(32,35);(5,9)(7,15)(10,16)(11,12)(14,19)(17,18)(20,21)(22,25)(23,26)(24,27)(28,32)(29,31)(30,33)(34,35);(4,6)(5,10)(7,14)(8,13)(9,16)(11,18)(12,17)(15,19)(20,22)(21,25)(23,28)(24,29)(26,32)(27,31)(30,34)(33,35);(6,13)(10,17)(14,19)(16,18)(20,23,24,30)(21,26,27,33)(22,28,29,34)(25,32,31,35)111 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 11229);