Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32944 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26 |
| direct product | C3×#32946 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C22×C3×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G | |
1294 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 39 points:
(1,2,3)(4,5)(6,7)(8,9,14,19)(10,20,22,31)(11,25,27,33)(12,18,13,17)(15,24,30,23)(16,28,32,26)(21,35,36,29)(34,38,39,37);(8,10,13,24,14,22,12,23)(9,15,18,31,19,30,17,20)(11,21,28,38,27,36,26,37)(16,29,33,39,32,35,25,34);(8,11)(9,16)(10,21)(12,26)(13,28)(14,27)(15,29)(17,25)(18,33)(19,32)(20,34)(22,36)(23,37)(24,38)(30,35)(31,39);(5,7)(10,22)(15,30)(20,31)(21,36)(23,24)(29,35)(34,39)(37,38)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 25934);