Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29748 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C23×C4 |
| direct product | C3×#29749 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C22×C3×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G | |
904 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)(6,9)(7,8)(10,11)(12,13,16,21,18,23,17,22)(14,19,25,34,26,35,32,39)(15,27,29,38,31,37,30,20)(24,36,28,42,40,43,41,33);(5,8)(9,11)(12,14)(13,19)(15,28)(16,25)(17,32)(18,26)(20,36)(21,34)(22,39)(23,35)(24,30)(27,42)(29,40)(31,41)(33,37)(38,43);(4,6)(5,9)(7,10)(8,11)(12,15,18,31)(13,20,23,38)(14,24,26,40)(16,30,17,29)(19,33,35,42)(21,37,22,27)(25,41,32,28)(34,43,39,36);(6,10)(9,11)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 22737);