Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10218 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C3×#10224 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #392 |
| commutator, Frattini, cosocle | C2×C42 | C23×C3 |
| prime radical | C23×C3 | #127 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
462 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2,3)(4,5,10,14)(6,15,7,18)(8,13,9,12)(11,17,19,16)(22,29)(23,24)(26,30)(27,33)(31,32)(34,35);(4,6,10,7)(5,11,14,19)(8,17,9,16)(12,18,13,15)(20,21)(22,30)(23,27)(24,33)(25,28)(26,32)(29,34)(31,35);(4,7,10,6)(5,11,14,19)(8,16,9,17)(12,18,13,15)(20,22)(21,26)(23,32)(24,29)(25,31)(27,35)(28,34)(30,33)102 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 3212);