Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#19417 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C24×C3 |
| prime radical | C2×C3×#41 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
608 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)(4,5,8,11)(6,12,15,19)(7,16,13,10)(9,17,18,14)(20,21,25,29)(22,26,31,35)(23,28,24,33)(27,32,34,30);(4,6,13,17,8,15,7,14)(5,9,16,19,11,18,10,12)(20,22)(21,26)(23,32)(24,30)(25,31)(27,33)(28,34)(29,35);(20,22)(21,27)(23,30)(24,32)(25,31)(26,33)(28,35)(29,34);(20,23,25,24)(21,28,29,33)(22,30,31,32)(26,27,35,34)147 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 12405);