Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C7 · Schur multiplier (H2) = C26 |
| direct product | C7×#20855 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | #120 |
| commutator | C2×C4 | C23×C4×C7 |
| Frattini, cosocle | C22×C4 | C24×C7 |
| socle | C22×C7 | C2×#120 |
| prime radical | C22×C7×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C23×C4×C7 ◁ G | |
1422 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)(8,9,11,10)(12,13)(14,21)(15,19)(16,24)(17,20)(18,22)(23,26)(25,27);(8,10,11,9)(12,14)(13,18)(15,22)(16,25)(17,23)(19,26)(20,27)(21,24);(12,13)(14,18)(15,19)(16,24)(17,20)(21,25)(22,26)(23,27);(12,15,17,16)(13,19,20,24)(14,22,23,25)(18,26,27,21)448 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 13843);