Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1619 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×#8449 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C7 | #120 |
| commutator | C42 | C22×C4×C7 |
| Frattini, cosocle | C2×C42 | C23×C7 |
| prime radical | C22×C7×D4 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C7 ◁ C2×C42×C7 ◁ G | |
788 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3,4,5,6,7)(8,9)(10,11)(13,14)(15,16)(18,19)(20,21,24,26,25,22,23,27);(9,11)(12,13,17,19)(14,15,18,16)(20,22)(21,25)(23,27)(24,26);(12,14)(13,15)(16,19)(17,18)(20,23,25,24)(21,26,22,27)364 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 1437);