Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23886 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#26403 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C2×C3×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4 ◁ G | |
706 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 43 points:
(1,2,3)(4,5,8,10)(6,9,7,11)(12,13)(14,25)(15,20)(16,33)(17,23)(18,24)(19,28)(21,32)(22,35)(26,39)(27,36)(29,37)(30,43)(31,40)(34,38)(41,42);(4,6,8,7)(5,9,10,11)(12,14)(13,19)(15,30)(16,27)(17,34)(18,29)(20,39)(21,36)(22,37)(23,25)(24,38)(26,32)(28,35)(31,42)(33,41)(40,43);(4,7,8,6)(5,9,10,11)(12,15,18,16)(13,20,24,33)(14,26,29,42)(17,31,35,32)(19,36,38,43)(21,23,40,22)(25,39,37,41)(27,34,30,28);(12,16,18,15)(13,21,24,40)(14,27,29,30)(17,31,35,32)(19,36,38,43)(20,23,33,22)(25,41,37,39)(26,34,42,28)129 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 19391);