Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3026 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C7 · Schur multiplier (H2) = C210 |
| direct product | C7×#60398 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | C2×#120 |
| commutator, Frattini, cosocle | C23 | C25×C7 |
| socle | C2×C7 | C22×#120 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C23×C4×C7 ◁ G | |
4836 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)(10,14)(11,15)(12,17)(13,16)(18,30)(19,26)(20,25)(21,28)(22,27)(23,29)(24,32)(31,38)(33,36)(34,37)(35,39);(8,10)(9,14)(11,18)(12,19)(13,20)(15,24)(16,25)(17,26)(21,31)(22,32)(23,33)(27,30)(28,35)(29,36)(34,39)(37,38);(8,11)(9,15)(10,18)(12,21)(13,22)(14,24)(16,27)(17,28)(19,31)(20,32)(23,34)(25,30)(26,35)(29,37)(33,39)(36,38);(9,16)(14,25)(15,27)(17,29)(24,30)(26,36)(28,37)(35,38);(8,12,13,23)(9,17,16,29)(10,19,20,33)(11,21,22,34)(14,26,25,36)(15,28,27,37)(18,31,32,39)(24,35,30,38)406 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 53386);