Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24274 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C23×C4 |
| direct product | C3×#27781 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C22×C3×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C3×C4×C4⋊C4 ◁ G | |
776 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 31 points:
(1,2,3)(5,6)(8,9)(10,12)(11,13)(14,15)(16,17,21,24,20,25,19,29)(18,26,27,31,28,22,30,23);(4,5,7,6)(8,10,11,14)(9,12,13,15)(16,18,19,30,20,28,21,27)(17,22,29,31,25,26,24,23);(9,13)(12,15)(16,18,20,28)(17,23,25,31)(19,27,21,30)(22,29,26,24);(4,6,7,5)(16,19,20,21)(17,24,25,29)(18,27,28,30)(22,31,26,23)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 20769);