Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1462 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C7 · Schur multiplier (H2) = C22×C4 |
| direct product | C7×#8157 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | C2×C22⋊C4 |
| commutator | C23 | C2×C42×C7 |
| Frattini, cosocle | C2×C42 | C23×C7 |
| socle | C22×C7 | C4×C22⋊C4 |
| prime radical | C24×C7 | C42 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C2×C4×C7×C8 ◁ G | |
482 conjugacy classes of subgroups.
| minimal degree | μ(G) = 47 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 3 generators on 47 points:
(1,2,3,4,5,6,7)(16,17,20,24,21,25,32,38)(18,26,29,40,28,22,33,35)(19,23,30,36,31,37,43,46)(27,39,42,47,41,34,44,45);(8,9,10,13,11,12,14,15)(16,18)(17,22,25,26)(19,27)(20,29)(21,28)(23,34,37,39)(24,35,38,40)(30,42)(31,41)(32,33)(36,45,46,47)(43,44);(9,12)(13,15)(16,19,20,30,21,31,32,43)(17,23,24,36,25,37,38,46)(18,27,29,42,28,41,33,44)(22,34,35,45,26,39,40,47)448 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 1145);