Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33758 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C23×C4 |
| direct product | C3×#33780 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C3 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ C2×C3×C4×C8 ◁ G | |
632 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(4,5)(6,10)(7,9)(8,11)(13,17)(14,20)(16,23)(18,26)(19,22)(21,27)(24,25);(4,6,7,11)(5,8,9,10)(12,13,15,17)(14,18,21,25)(16,19,22,23)(20,24,27,26);(6,11)(8,10)(12,14)(13,18)(15,21)(16,24)(17,25)(19,27)(20,23)(22,26);(5,9)(8,10)(12,13,15,17)(14,18,21,25)(16,19,22,23)(20,24,27,26)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26768);