Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3585 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C7 · Schur multiplier (H2) = C29 |
| direct product | C7×#60184 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | C23×D4 |
| commutator | C4 | C24×C4×C7 |
| Frattini, cosocle | C2×C4 | C25×C7 |
| socle | C2×C7 | C22×C4×D4 |
| prime radical | C7×#3584 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C2×C7×#131 ◁ G | |
2604 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 5 generators on 39 points:
(1,2,3,4,5,6,7)(8,9,13,18,14,10,19,17)(11,20,23,29,24,15,27,30)(12,16,26,32,25,21,33,31)(22,34,36,38,35,28,37,39);(8,10)(9,14)(11,20)(12,21)(13,17)(15,24)(16,25)(18,19)(22,34)(23,29)(26,31)(27,30)(28,35)(32,33)(36,38)(37,39);(8,11,13,23,14,24,19,27)(9,15,18,30,10,20,17,29)(12,22,26,36,25,35,33,37)(16,28,32,39,21,34,31,38);(8,12)(9,16)(10,21)(11,22)(13,26)(14,25)(15,28)(17,31)(18,32)(19,33)(20,34)(23,36)(24,35)(27,37)(29,38)(30,39);(12,25)(16,21)(22,35)(26,33)(28,34)(31,32)(36,37)(38,39)574 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 53172);