Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #20671 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×#33426 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | C2×#117 |
| commutator | C2×C4 | C23×C4×C7 |
| Frattini, cosocle | C2×C8 | C24×C7 |
| socle | C2×C7 | C2×#432 |
| prime radical | C2×C7×#41 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C2×C7×#97 ◁ G | |
520 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 4 generators on 39 points:
(1,2,3,4,5,6,7)(8,9,11,17,12,18,25,34,13,16,24,32,26,15,30,31)(10,19,21,33,22,27,36,38,20,14,23,28,35,29,37,39);(8,10)(9,14)(11,23)(12,22)(13,20)(15,27)(16,19)(17,33)(18,29)(21,24)(25,37)(26,35)(28,32)(30,36)(31,39)(34,38);(9,15,16,18)(11,24)(14,27,19,29)(17,34,32,31)(21,23)(25,30)(28,39,33,38)(36,37);(10,20)(14,19)(21,23)(22,35)(27,29)(28,33)(36,37)(38,39)406 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 26414);