Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C24 |
| direct product | C3×#17897 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C22×D4 |
| commutator | C2×C4 | C23×C3×C4 |
| Frattini, cosocle | C42 | C24×C3 |
| socle | C22×C3 | C2×C4×D4 |
| prime radical | C23×C3 | C4×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C3×#431 ◁ G | |
614 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 43 points:
(1,2,3)(4,5,8,6,7,9,11,10)(12,13)(14,22)(15,19)(16,28)(17,20)(18,24)(21,30)(23,36)(25,35)(26,34)(27,40)(29,32)(31,38)(33,39)(37,43)(41,42);(5,9)(6,10)(12,14,17,25)(13,18,20,32)(15,23,26,37)(16,29,30,24)(19,31,34,42)(21,35,28,22)(27,41,39,38)(33,43,40,36);(4,6,8,9,7,10,11,5)(12,15,17,26)(13,19,20,34)(14,23,25,37)(16,27,30,39)(18,31,32,42)(21,33,28,40)(22,36,35,43)(24,38,29,41);(13,20)(15,26)(18,32)(21,28)(22,35)(23,37)(27,39)(38,41)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 10885);