Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1795 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C7 · Schur multiplier (H2) = C26 |
| direct product | C7×#10776 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | #379 |
| commutator | C2×C4 | C2×C42×C7 |
| Frattini, cosocle | C23×C4 | C23×C7 |
| prime radical | C23×C7×D4 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C23×C4×C7 ◁ G | |
1394 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3,4,5,6,7)(12,13,16,19,17,20,15,21)(14,22,23,26,24,18,25,27);(8,9,10,11)(12,14)(13,18)(15,23)(16,25)(17,24)(19,26)(20,22)(21,27);(13,19)(15,16)(18,26)(20,21)(22,27)(23,25)406 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 3764);