Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18316 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C7 · Schur multiplier (H2) = C24×C4 |
| direct product | C7×#18334 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C7 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G | |
972 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 4 generators on 39 points:
(1,2,3,4,5,6,7)(9,15)(10,20)(11,24)(12,21)(16,30)(17,34)(18,27)(19,32)(22,26)(23,25)(28,37)(31,33)(35,38)(36,39);(8,9)(10,16)(11,17)(12,27)(13,19)(14,18)(15,21)(20,31)(22,35)(23,34)(24,33)(25,38)(26,30)(28,39)(29,36)(32,37);(8,10,14,23)(9,16,18,34)(11,21,26,12)(13,20,29,25)(15,30,27,17)(19,31,36,38)(22,37,24,28)(32,33,39,35);(8,11,13,22)(9,17,19,35)(10,21,20,37)(12,25,28,23)(14,26,29,24)(15,31,32,16)(18,30,36,33)(27,38,39,34)259 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 11322);