Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24390 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C22×C4 |
| direct product | C3×#29174 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C2×C3×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C2×C3×#117 ◁ G | |
784 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)(7,9)(8,11)(12,13,15,17)(14,16,18,19)(22,28)(23,31)(25,32)(27,33)(29,34)(30,35);(4,6,7,11)(5,8,9,10)(12,14,15,18)(13,16,17,19)(20,21)(22,29)(23,27)(24,26)(25,30)(28,34)(31,33)(32,35);(14,18)(16,19)(20,22,23,28,24,30,31,35)(21,25,27,32,26,29,33,34);(5,9)(8,10)(21,26)(25,29)(27,33)(32,34)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 22162);