Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2010 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C27 |
| direct product | C2×C3×#2017 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | #127 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| socle | C22×C3 | #392 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C23×C3×C4 ◁ G | |
1938 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)(5,14)(6,19)(7,24)(8,12)(9,13)(10,16)(11,22)(15,32)(17,27)(18,26)(20,31)(21,30)(23,29)(25,33)(28,34);(3,5)(4,10)(6,20)(7,17)(8,16)(9,18)(11,28)(12,26)(13,27)(14,24)(15,21)(19,25)(22,33)(23,34)(29,32)(30,31);(3,6)(4,11)(5,15)(7,21)(8,22)(9,23)(10,25)(12,29)(13,30)(14,31)(16,20)(17,34)(18,33)(19,24)(26,28)(27,32)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 16421);