Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3055 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×#17963 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | C22×D4 |
| commutator | C2×C4 | C23×C4×C7 |
| Frattini, cosocle | C42 | C24×C7 |
| socle | C22×C7 | C2×C4×D4 |
| prime radical | C7×#466 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C2×C4×C7×C8 ◁ G | |
614 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 4 generators on 39 points:
(1,2,3,4,5,6,7)(8,9)(10,17)(11,16)(12,20)(13,15)(14,18)(19,23)(21,22)(24,25,29,32,30,26,28,33)(27,34,36,39,37,31,35,38);(8,10)(9,14)(11,18)(12,21)(13,19)(15,22)(16,23)(17,20)(24,26)(25,30)(27,34)(28,32)(29,33)(31,37)(35,38)(36,39);(9,15)(14,22)(16,20)(17,23)(24,27,29,36,30,37,28,35)(25,31,32,38,26,34,33,39);(8,11,13,12)(9,16,15,20)(10,18,19,21)(14,23,22,17)(24,28,30,29)(25,32,26,33)(27,35,37,36)(31,38,34,39)448 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 10951);