Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1595 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C2×C3×#1617 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | #120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C24×C3 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C23×C3×C4 ◁ G | |
984 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2)(35,36,37);(3,4,9,24)(5,14,16,31)(6,19,25,21)(7,12,8,13)(10,26,27,15)(11,23,34,22)(17,30,18,32)(20,28,33,29);(3,5,7,17)(4,10,12,28)(6,20,22,15)(8,18,9,16)(11,30,19,14)(13,29,24,27)(21,31,34,32)(23,26,25,33);(3,4,8,13)(5,10,18,29)(6,21,23,11)(7,12,9,24)(14,15,32,33)(16,27,17,28)(19,22,34,25)(20,31,26,30)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 11106);