Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19651 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C7 · Schur multiplier (H2) = C23×C4 |
| direct product | C7×#19665 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C2×#127 |
| commutator, Frattini, cosocle | C42 | C24×C7 |
| prime radical | C7×#3207 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G | |
712 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 4 generators on 39 points:
(1,2,3,4,5,6,7)(9,13)(11,18)(14,19)(15,20)(17,22)(21,23)(24,25)(26,33)(27,31)(28,37)(29,32)(30,35)(34,38)(36,39);(8,9,11,13,12,15,18,20)(10,14,17,19,16,21,22,23);(8,10)(9,14)(11,17)(12,16)(13,19)(15,21)(18,22)(20,23)(24,26,28,35,29,36,27,34)(25,30,31,33,32,38,37,39);(10,16)(14,21)(17,22)(19,23)(24,27,29,28)(25,31,32,37)(26,34,36,35)(30,33,38,39)364 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 12653);