Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60372 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C28 |
| direct product | C3×#60581 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C3 |
| prime radical | C3×#3005 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×#117 ◁ G | |
2024 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 43 points:
(1,2,3)(4,5)(6,9)(7,8)(10,11)(12,13,17,22)(14,18,25,32)(15,20,26,34)(16,19,28,21)(23,30,36,41)(24,29,38,31)(27,33,39,35)(37,40,43,42);(5,8)(9,11)(12,14,17,25)(13,18,22,32)(15,23,26,36)(16,24,28,38)(19,29,21,31)(20,30,34,41)(27,37,39,43)(33,40,35,42);(13,19)(16,28)(18,29)(20,33)(21,22)(24,38)(27,39)(30,40)(31,32)(34,35)(37,43)(41,42);(4,6,7,10)(5,9,8,11)(12,15)(13,20)(14,23)(16,27)(17,26)(18,30)(19,33)(21,35)(22,34)(24,37)(25,36)(28,39)(29,40)(31,42)(32,41)(38,43);(15,26)(20,34)(23,36)(27,39)(30,41)(33,35)(37,43)(40,42)183 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 53569);