Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13334 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×#13334 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | #2011 |
| commutator, Frattini, cosocle | C22×C8 | C23×C7 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C22×C4×C7 ◁ C7×#405 ◁ G | |
568 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3,4,5,6,7)(8,9)(10,21)(11,15)(12,28)(13,31)(14,19)(16,22)(17,25)(18,29)(20,27)(23,36)(24,33)(26,35)(30,38)(32,37)(34,39);(9,15)(11,25)(12,29)(13,27)(16,21)(17,20)(18,31)(19,28)(22,30)(23,32)(26,39)(33,34)(35,36)(37,38);(8,10)(9,16)(11,26)(12,30)(13,32)(14,24)(15,33)(17,36)(18,37)(19,34)(20,38)(21,28)(22,29)(23,27)(25,39)(31,35)238 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6322);