Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23886 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#25984 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C2×C3×H(5,𝔽2) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4 ◁ G | |
876 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(4,5,7,9)(6,10,11,8)(12,13)(14,17)(15,22)(16,18)(19,23)(20,26)(21,25)(24,27);(13,18)(14,20)(16,26)(19,22)(21,27)(23,25);(4,6,7,11)(5,8,9,10)(12,14,17,13)(15,21,24,19)(16,18,26,20)(22,23,27,25);(12,15)(13,19)(14,21)(16,25)(17,24)(18,22)(20,27)(23,26)129 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 18972);