Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60399 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C7 · Schur multiplier (H2) = C210 |
| direct product | C7×#60401 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | C22×#120 |
| commutator, Frattini, cosocle | C23 | C25×C7 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C22×C7×D4 ◁ G | |
4854 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)(8,9,13,17)(10,14,20,26)(11,15,22,28)(12,16,23,29)(18,30,32,24)(19,25,33,36)(21,27,34,37)(31,38,39,35);(8,10)(9,14)(11,18)(12,19)(13,20)(15,24)(16,25)(17,26)(21,31)(22,32)(23,33)(27,35)(28,30)(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)(25,35)(26,30)(29,37)(33,39)(36,38);(8,12)(9,16)(10,19)(11,21)(13,23)(14,25)(15,27)(17,29)(18,31)(20,33)(22,34)(24,35)(26,36)(28,37)(30,38)(32,39);(12,23)(16,29)(19,33)(21,34)(25,36)(27,37)(31,39)(35,38)406 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 53389);