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×#1615 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | #120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C2×C3×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C23×C3×C4 ◁ G | |
1016 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);(3,4,10,20)(5,14,25,34)(6,8,13,9)(7,19,26,30)(11,27,33,16)(12,23,22,24)(15,31,28,21)(17,29,18,32);(3,5)(4,11)(6,15)(7,21)(8,18)(9,17)(10,25)(12,29)(13,28)(14,19)(16,24)(20,33)(22,32)(23,27)(26,31)(30,34);(3,6,9,4)(5,15,17,11)(7,22,24,30)(8,20,10,13)(12,23,19,26)(14,31,29,27)(16,34,21,32)(18,33,25,28)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 11094);