Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24398 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#24624 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C24×C3 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G | |
670 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)(6,11)(7,9)(8,10)(12,13,16,21,18,23,17,22)(14,19,25,34,26,35,32,39)(15,27,30,38,31,37,29,20)(24,36,28,42,41,43,40,33);(4,6,8,7)(5,9,10,11)(12,14)(13,19)(15,28)(16,25)(17,32)(18,26)(20,36)(21,34)(22,39)(23,35)(24,29)(27,42)(30,41)(31,40)(33,37)(38,43);(4,7,8,6)(5,9,10,11)(12,15)(13,20)(14,24)(16,29)(17,30)(18,31)(19,33)(21,37)(22,27)(23,38)(25,40)(26,41)(28,32)(34,43)(35,42)(36,39);(5,10)(9,11)(12,16,18,17)(13,21,23,22)(14,25,26,32)(15,29,31,30)(19,34,35,39)(20,37,38,27)(24,40,41,28)(33,43,42,36)129 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 17612);