Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2008 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C27 |
| direct product | C2×C3×#2009 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #392 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C2×C3×#575 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G | |
1926 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)(5,15)(6,19)(7,13)(8,12)(9,14)(10,17)(11,21)(16,24)(18,29)(20,28)(22,27)(23,25)(26,32)(30,31)(33,34);(3,5)(4,10)(6,16)(7,23)(8,17)(9,18)(11,26)(12,29)(13,15)(14,25)(19,30)(20,33)(21,31)(22,32)(24,28)(27,34);(3,6,9,22)(4,11,14,28)(5,16,18,32)(7,21,8,20)(10,24,25,26)(12,19,13,27)(15,30,29,34)(17,33,23,31)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 16399);