Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23400 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C27 |
| direct product | C3×#23408 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C3×#3449 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G | |
1946 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)(8,9,14,19)(10,20,22,32)(11,24,26,33)(12,18,13,17)(15,27,30,23)(16,28,31,25)(21,29,35,39)(34,36,38,37);(4,5)(6,7)(8,10)(9,15)(11,21)(12,27)(13,23)(14,22)(16,29)(17,32)(18,20)(19,30)(24,38)(25,37)(26,35)(28,36)(31,39)(33,34);(5,7)(8,11)(9,16)(10,21)(12,25)(13,28)(14,26)(15,29)(17,24)(18,33)(19,31)(20,34)(22,35)(23,36)(27,37)(30,39)(32,38);(10,22)(15,30)(20,32)(21,35)(23,27)(29,39)(34,38)(36,37)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 16396);