Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #14715 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#15139 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24×C3 | C24 |
| commutator, Frattini, cosocle | C24 | C24×C3 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24×C3 ◁ G | |
1250 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 768 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(5,7)(12,13,17,15)(14,19,16,18)(22,27)(23,25);(9,11)(12,14,17,16)(13,18,15,19)(20,21,24,26)(22,25,27,23);(8,9)(10,11)(12,15,17,13)(14,19,16,18)(20,22)(21,25)(23,26)(24,27);(4,5,6,7)(12,16,17,14)(13,19,15,18)(20,23,24,25)(21,22,26,27)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 8127);