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×#27673 |
| 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)(14,26)(15,30)(16,20)(17,24)(18,23)(19,25)(21,32)(22,29)(27,41)(28,36)(31,40)(33,39)(34,38)(35,42)(37,43);(4,6)(5,8)(7,10)(9,11)(12,14,18,28,19,29,17,16)(13,20,24,22,25,36,23,26)(15,27,34,43,33,42,32,31)(21,35,39,37,38,41,30,40);(6,10)(8,11)(12,15)(13,21)(14,27)(16,31)(17,32)(18,34)(19,33)(20,35)(22,37)(23,30)(24,39)(25,38)(26,40)(28,43)(29,42)(36,41);(12,16,17,29,19,28,18,14)(13,22,23,20,25,26,24,36)(15,31,32,42,33,43,34,27)(21,37,30,35,38,40,39,41)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 20661);