Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42×C7 · Schur multiplier (H2) = C27 |
| direct product | C7×#20351 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | C22×D4 |
| commutator | C4 | C22×C42×C7 |
| Frattini, cosocle | C22×C4 | C24×C7 |
| socle | C22×C7 | C22×C4⋊C4 |
| prime radical | C22×C7×#41 | C22 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C22×C7×#41 ◁ G | |
1978 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 4 generators on 43 points:
(1,2,3,4,5,6,7)(8,9)(10,11)(12,13,16,22,18,23,17,21)(14,24,26,19,28,33,27,34)(15,20,30,37,29,35,31,36)(25,38,40,32,39,43,41,42);(9,11)(12,14,17,27,18,28,16,26)(13,19,22,33,23,34,21,24)(15,25,31,41,29,39,30,40)(20,32,37,43,35,42,36,38);(12,15)(13,20)(14,25)(16,30)(17,31)(18,29)(19,32)(21,36)(22,37)(23,35)(24,38)(26,40)(27,41)(28,39)(33,43)(34,42);(15,29)(20,35)(25,39)(30,31)(32,42)(36,37)(38,43)(40,41)616 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 13339);