Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2002 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C7 · Schur multiplier (H2) = C26 |
| direct product | C7×#23498 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | #120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C7 |
| socle | C22×C7 | C2×#120 |
| prime radical | C7×#3449 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C23×C4×C7 ◁ G | |
1450 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3,4,5,6,7)(9,10)(12,13)(14,15)(16,20)(17,21)(18,22)(19,26)(23,25)(24,27);(8,9)(10,11)(12,14)(13,15)(16,23)(17,19)(18,24)(20,25)(21,26)(22,27);(8,10,11,9)(12,15)(13,19)(14,20)(16,25)(17,26)(18,27)(21,24)(22,23);(12,16,18,17)(13,20,22,21)(14,23,24,19)(15,25,27,26)364 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 16486);