Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13307 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C3×#13385 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #447 |
| commutator, Frattini, cosocle | C22×C8 | C23×C3 |
| prime radical | C3×#3061 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C2×C3×#131 ◁ G | |
408 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3)(5,6)(8,9)(10,20)(11,16)(12,15)(13,18)(14,22)(17,27)(19,26)(21,29)(23,37)(24,33)(25,31)(28,38)(30,36)(32,39)(34,35);(4,5)(6,7)(8,10)(9,14)(11,22)(12,23)(13,21)(15,30)(16,20)(17,28)(18,33)(19,29)(24,26)(25,36)(27,35)(31,37)(32,34)(38,39);(4,6,7,5)(9,15)(10,21)(12,27)(13,26)(14,28)(16,31)(17,19)(18,32)(20,34)(22,24)(23,36)(25,39)(29,37)(30,33)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 6373);