Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30441 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#30442 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | #2668 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C3×#3021 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C22×C3×C4 ◁ G | |
878 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2,3)(4,5,10,15)(6,16,20,27)(7,21,23,13)(8,11,9,14)(12,17,28,19)(18,30,33,35)(22,25,24,29)(26,31,34,32);(5,11)(6,17)(7,22)(8,9)(13,21)(14,15)(16,27)(19,20)(23,24)(26,30)(31,32)(34,35);(4,6)(5,12)(7,18)(8,17)(9,19)(10,20)(11,16)(13,26)(14,27)(15,28)(21,34)(22,32)(23,33)(24,31)(25,30)(29,35);(4,7)(5,13)(6,18)(8,24)(9,22)(10,23)(11,25)(12,26)(14,29)(15,21)(16,30)(17,31)(19,32)(20,33)(27,35)(28,34)102 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 23430);