Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1902 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C42 · Schur multiplier (H2) = C23×C4 |
| direct product | C3×#10905 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | #386 |
| commutator | C2×C4 | C2×C3×C42 |
| Frattini, cosocle | C23×C4 | C23×C3 |
| socle | C2×C3 | #1436 |
| prime radical | C24×C3 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C23×C3×C4 ◁ G | |
672 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2,3)(4,5,9,13,10,14,8,23)(6,15,18,27,19,26,17,11)(7,20,21,12,22,28,24,29)(16,31,32,35,34,25,33,30);(4,6)(5,11,14,27)(7,16)(8,17)(9,18)(10,19)(12,25,29,31)(13,15,23,26)(20,35,28,30)(21,32)(22,34)(24,33);(4,7,9,21,10,22,8,24)(5,12,13,28,14,29,23,20)(6,16,18,32,19,34,17,33)(11,25,15,30,27,31,26,35)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 3893);