Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3192 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C28 |
| direct product | C2×C3×#3195 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C3 |
| prime radical | C25×C3 | D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C24×C3×C4 ◁ G | |
2322 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 29 points:
(1,2)(27,28,29);(3,4)(5,9)(6,8)(7,10)(11,12,15,18)(13,19,20,26)(14,22,24,17)(16,23,25,21);(3,5)(4,7)(6,10)(8,9)(13,20)(16,25)(19,26)(21,23);(5,10)(7,9)(11,13,15,20)(12,16,18,25)(14,23,24,21)(17,26,22,19);(4,8)(7,9)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 53726);