Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24278 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#27674 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C3×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G | |
696 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)(4,5,7,9)(6,8,10,11)(12,13)(14,22)(15,29)(16,33)(17,24)(18,23)(19,25)(20,27)(21,31)(26,40)(28,36)(30,43)(32,39)(34,38)(35,42)(37,41);(4,6)(5,8)(7,10)(9,11)(12,14,17,27,19,28,18,16)(13,20,23,22,25,33,24,36)(15,26,31,41,32,42,34,30)(21,35,29,37,38,40,39,43);(6,10)(8,11)(12,15)(13,21)(14,26)(16,30)(17,31)(18,34)(19,32)(20,35)(22,37)(23,29)(24,39)(25,38)(27,41)(28,42)(33,40)(36,43);(12,16,18,28,19,27,17,14)(13,22,24,20,25,36,23,33)(15,30,34,42,32,41,31,26)(21,37,39,35,38,43,29,40)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 20662);