Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2010 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C26 |
| direct product | C3×#11040 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | #127 |
| commutator | C22×C4 | C22×C3×C4 |
| Frattini, cosocle | C23×C4 | C23×C3 |
| socle | C22×C3 | #392 |
| prime radical | C2×C3×#575 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C23×C3×C4 ◁ G | |
1278 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)(4,5,6,7)(8,9)(10,19)(11,24)(12,29)(13,17)(14,18)(15,21)(16,27)(20,37)(22,32)(23,31)(25,36)(26,35)(28,34)(30,38)(33,39);(8,10)(9,15)(11,25)(12,22)(13,21)(14,23)(16,33)(17,31)(18,32)(19,29)(20,26)(24,30)(27,38)(28,39)(34,37)(35,36);(8,11)(9,16)(10,20)(12,26)(13,27)(14,28)(15,30)(17,34)(18,35)(19,36)(21,25)(22,39)(23,38)(24,29)(31,33)(32,37)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 4028);