Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23 |
| direct product | C3×#9671 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | #121 |
| commutator | C22×C4 | C22×C3×C4 |
| Frattini, cosocle | C2×C42 | C23×C3 |
| socle | C22×C3 | #390 |
| prime radical | C23×C3 | C4×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C2×C3×C42 ◁ G | |
434 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 3 generators on 43 points:
(1,2,3)(4,5,9,10,8,6,7,11)(12,13)(14,18,23,32)(15,25)(16,28)(17,20)(19,26)(21,29)(22,34,35,43)(24,38,37,31)(27,39)(30,41,42,36)(33,40);(4,6,9,11,8,5,7,10)(12,14,17,23)(13,18)(15,22,27,35)(16,24,29,37)(19,30)(20,32)(21,31)(25,34)(26,36,40,41)(28,38)(33,42)(39,43);(4,7,8,9)(5,10,6,11)(12,15,16,26,17,27,29,40)(13,19,21,25,20,33,28,39)(14,22,24,36,23,35,37,41)(18,30,31,34,32,42,38,43)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 2659);