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) = C23×C4 |
| direct product | C7×#20788 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | C2×#117 |
| commutator | C2×C4 | C23×C4×C7 |
| Frattini, cosocle | C22×C4 | C24×C7 |
| socle | C2×C7 | #2283 |
| prime radical | C2×C7×#41 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C7×#405 ◁ G | |
608 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)(10,19,22,15)(11,16)(12,17)(13,18)(14,21,28,36)(20,34,23,29)(24,31)(25,32)(26,33)(27,35,30,37)(38,39);(8,10)(9,14,18,28)(11,23)(12,21)(13,22)(15,17,19,33)(16,30,32,27)(20,25)(24,37)(26,36)(29,39,34,31)(35,38);(8,10)(9,15)(11,20)(12,21)(13,22)(14,17)(16,29)(18,19)(23,25)(24,35)(26,36)(27,31)(28,33)(30,39)(32,34)(37,38);(8,11,12,24,13,25,26,38)(9,16,17,31,18,32,33,39)(10,20,21,35,22,23,36,37)(14,27,19,34,28,30,15,29)406 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 13776);