Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C26 |
| direct product | C3×#11038 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | #124 |
| commutator | C22×C4 | C22×C3×C4 |
| Frattini, cosocle | C23×C4 | C23×C3 |
| socle | C22×C3 | #394 |
| 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)(10,20)(11,25,28,16)(12,19)(13,17)(14,18)(15,22)(21,33,36,38)(23,31)(24,32)(26,30,37,39)(27,34,29,35);(5,7)(8,10)(9,15)(11,26)(12,24)(13,22)(14,23)(16,33)(17,31)(18,32)(19,20)(21,27)(25,38)(28,37)(29,36)(30,34)(35,39);(8,11)(9,16)(10,21)(12,27)(13,29)(14,28)(15,30)(17,34)(18,25)(19,35)(20,33)(22,26)(23,36)(24,37)(31,38)(32,39)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 4026);