Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23912 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C27 |
| direct product | C3×#23917 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G | |
1682 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)(10,15)(11,23)(12,17)(13,28)(14,19)(16,26)(18,21)(20,33)(22,32)(24,39)(25,35)(27,31)(29,34)(30,37)(36,38);(4,5)(6,7)(8,10,14,22)(9,15,19,32)(11,20,25,36)(12,13,27,21)(16,30,34,39)(17,28,31,18)(23,38,35,33)(24,29,37,26);(5,7)(8,11)(9,16)(10,20)(12,24)(13,29)(14,25)(15,30)(17,33)(18,35)(19,34)(21,26)(22,36)(23,28)(27,37)(31,38)(32,39);(8,12,14,27)(9,17,19,31)(10,21,22,13)(11,24,25,37)(15,18,32,28)(16,33,34,38)(20,26,36,29)(23,30,35,39)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 16905);