Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C7 · Schur multiplier (H2) = C22×C4 |
| direct product | C7×#7894 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C7 | C2×D4 |
| commutator | C2×C4 | C2×C42×C7 |
| Frattini, cosocle | C2×C42 | C23×C7 |
| socle | C23×C7 | C2×C22⋊C4 |
| prime radical | C24×C7 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C7 ◁ C22×C42×C7 ◁ G | |
614 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 3 generators on 43 points:
(1,2,3,4,5,6,7)(8,9)(10,11)(12,13)(14,22)(15,19)(16,20)(17,21)(18,23)(24,27)(25,26)(28,29,32,36,33,37,31,35)(30,38,39,42,41,43,40,34);(9,11)(12,14,16,24,17,25,15,23)(13,18,19,26,21,27,20,22)(28,30)(29,34,37,42)(31,40)(32,39)(33,41)(35,43,36,38);(12,15,17,16)(13,19,21,20)(14,23,25,24)(18,26,27,22)(28,31,33,32)(29,35,37,36)(30,39,41,40)(34,38,42,43)532 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 882);