Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C7 · Schur multiplier (H2) = C29 |
| direct product | C7×#61038 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C7 |
| socle | C22×C7 | C23×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C2×C4×C7×D4 ◁ G | |
2790 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 5 generators on 27 points:
(1,2,3,4,5,6,7)(8,9)(10,11)(12,13)(14,21)(15,19)(16,25)(17,20)(18,24)(22,23)(26,27);(12,14)(13,18)(15,24)(16,26)(17,22)(19,23)(20,27)(21,25);(9,11)(14,22)(18,27)(21,23)(24,26);(12,13)(14,23)(15,19)(16,25)(17,20)(18,26)(21,22)(24,27);(12,15,17,16)(13,19,20,25)(14,24,22,26)(18,23,27,21)406 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 54026);