Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24390 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C23 |
| direct product | C3×#29470 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C24×C3 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C2×C3×#117 ◁ G | |
700 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)(6,10)(7,8)(9,11)(12,13)(14,18)(15,17)(16,19)(20,21,25,30,26,31,24,29)(22,27,32,28,23,33,34,35);(5,8)(9,10)(12,14)(13,16)(15,19)(17,18)(20,22)(21,27)(23,26)(24,34)(25,32)(28,30)(29,35)(31,33);(14,19)(16,18)(20,23,26,22)(21,28,31,35)(24,34,25,32)(27,29,33,30);(4,6,7,11)(5,9,8,10)(13,17)(16,18)(20,24,26,25)(21,29,31,30)(22,32,23,34)(27,28,33,35)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 22458);