Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24404 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C7 · Schur multiplier (H2) = C25×C4 |
| direct product | C7×#24406 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C2×#120 |
| commutator, Frattini, cosocle | C24 | C24×C7 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C23×C4×C7 ◁ G | |
1620 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)(12,13)(14,20,22,17)(15,23,24,18)(16,19)(21,25)(26,27);(8,9)(10,11)(12,14,16,22)(13,17,19,20)(15,21,24,27)(18,25,23,26);(9,11)(12,13)(14,17)(15,18)(16,19)(20,22)(21,25)(23,24)(26,27);(12,15)(13,18)(14,21)(16,24)(17,25)(19,23)(20,26)(22,27)301 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 17394);