Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19713 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C7 · Schur multiplier (H2) = C23×C8 |
| direct product | C7×#19724 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C7 | C2×#127 |
| commutator, Frattini, cosocle | C42 | C24×C7 |
| prime radical | C7×#3184 | 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)(8,9)(10,18)(11,15)(12,16)(13,23)(14,17)(19,21)(20,22)(24,25)(26,32)(27,37)(28,33)(29,38)(30,34)(31,35)(36,39);(8,10,13,15,14,20,12,19)(9,11,16,18,17,21,23,22)(24,26,29,35,30,36,28,27)(25,31,33,32,34,37,38,39);(8,11,14,21)(9,15,17,19)(10,16,20,23)(12,22,13,18)(24,27,28,36,30,35,29,26)(25,32,38,31,34,39,33,37);(8,12,14,13)(9,16,17,23)(10,19,20,15)(11,18,21,22)(24,28,30,29)(25,33,34,38)(26,35,36,27)(31,39,37,32)280 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 12712);