Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23912 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C7 · Schur multiplier (H2) = C26 |
| direct product | C7×#24226 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C7 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G | |
1494 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)(9,10)(12,13)(14,22)(15,21)(16,20)(17,19)(18,24)(23,27)(25,26);(8,9,11,10)(12,14,16,24)(13,18,20,22)(15,23,17,25)(19,27,21,26);(13,19)(15,17)(18,26)(20,21)(22,27)(23,25);(8,10,11,9)(12,15,16,17)(13,19,20,21)(14,23,24,25)(18,27,22,26)280 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 17214);