Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2002 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C25 |
| direct product | C3×#11065 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | #120 |
| commutator | C22×C4 | C22×C3×C4 |
| Frattini, cosocle | C23×C4 | C23×C3 |
| socle | C22×C3 | #386 |
| prime radical | C22×C3×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C23×C3×C4 ◁ G | |
870 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3)(5,7)(8,9,13,17,14,19,12,18)(10,15,22,30,23,31,21,35)(11,24,26,16,27,33,28,34)(20,32,36,29,37,39,25,38);(4,5,6,7)(8,10)(9,15)(11,25)(12,21)(13,22)(14,23)(16,32)(17,30)(18,35)(19,31)(20,26)(24,38)(27,36)(28,37)(29,33)(34,39);(8,11)(9,16)(10,20)(12,28)(13,26)(14,27)(15,29)(17,33)(18,24)(19,34)(21,25)(22,36)(23,37)(30,39)(31,38)(32,35)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 4053);