Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1655 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C22 |
| direct product | C7×#8917 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | #120 |
| commutator | C42 | C22×C4×C7 |
| Frattini, cosocle | C2×C42 | C23×C7 |
| socle | C22×C7 | #385 |
| prime radical | C7×#121 | Q8 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C2×C42×C7 ◁ G | |
338 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3,4,5,6,7)(8,9,13,17,14,18,12,20)(10,19,16,15,22,23,21,11)(24,25,28,31)(26,32,33,39)(27,35,36,30)(29,37,38,34);(8,10,13,16,14,22,12,21)(9,15,17,23,18,11,20,19)(27,36)(30,35)(32,39)(34,37);(8,11)(9,16)(10,20)(12,23)(13,19)(14,15)(17,22)(18,21)(24,26)(25,29)(27,37)(28,33)(30,39)(31,38)(32,35)(34,36)280 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 1905);