Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×C3×#3137 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C3 |
| prime radical | C22×C3×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C2×C3×#117 ◁ G | |
2070 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 5 generators on 29 points:
(1,2)(27,28,29);(3,4)(5,8)(6,7)(9,10)(11,12,16,18)(13,19,20,26)(14,15,22,23)(17,24,25,21);(4,7)(8,10)(11,13,15,21,16,20,23,24)(12,17,14,19,18,25,22,26);(3,5,6,9)(4,8,7,10)(11,14,15,18,16,22,23,12)(13,19,21,25,20,26,24,17);(13,20)(17,25)(19,26)(21,24)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 53599);