Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7654 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C4 |
| direct product | C3×#7657 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C3 | C2×C22⋊C4 |
| commutator | C22×C4 | C22×C3×C4 |
| Frattini, cosocle | C2×C42 | C23×C3 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C3×C43 ◁ G | |
370 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 768 |
A minimal faithful permutation representation — 3 generators on 43 points:
(1,2,3)(5,8,9,10)(12,13)(14,20,22,17)(15,18)(16,19)(21,26,27,24)(23,25)(28,29,31,34,32,35,38,41)(30,33,37,40,36,39,42,43);(4,5)(6,10)(7,9)(8,11)(12,14,16,22)(13,17,19,20)(15,21,23,27)(18,24,25,26)(28,30)(29,33)(31,37)(32,36)(34,40)(35,39)(38,42)(41,43);(12,15,16,23)(13,18,19,25)(14,21,22,27)(17,24,20,26)(30,36)(33,39)(37,42)(40,43)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 645);