Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C7 · Schur multiplier (H2) = C23×C4 |
| direct product | C7×#20787 |
| subgroup | G/H | |
|---|---|---|
| center | C7×C8 | #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 ◁ C7×C8 ◁ C7×#405 ◁ G | |
616 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)(14,21,27,35)(20,32,34,28)(23,29)(24,30)(25,31)(26,33,37,39)(36,38);(8,10)(9,14,18,27)(11,20)(12,21)(13,22)(15,17,19,31)(16,26,30,37)(23,33)(24,34)(25,35)(28,29,32,38)(36,39);(8,10)(9,15)(11,20)(12,21)(13,22)(14,17)(16,28)(18,19)(23,33)(24,34)(25,35)(26,29)(27,31)(30,32)(36,39)(37,38);(8,11,12,23,13,24,25,36)(9,16,17,29,18,30,31,38)(10,20,21,33,22,34,35,39)(14,26,19,32,27,37,15,28)448 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 13775);