Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1714 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C7 · Schur multiplier (H2) = C23×C4 |
| direct product | C7×#19613 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | #127 |
| commutator, Frattini, cosocle | C42 | C24×C7 |
| socle | C22×C7 | C2×#127 |
| prime radical | C7×#3169 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C2×C42×C7 ◁ G | |
734 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)(10,17)(11,15)(12,21)(13,16)(14,20)(18,23)(19,22)(24,25)(26,32)(27,29)(28,36)(30,34)(31,35)(33,38)(37,39);(8,10,12,20,13,18,11,19)(9,14,15,17,16,22,21,23)(25,29)(30,37)(31,36)(32,38);(10,18)(14,22)(17,23)(19,20)(24,26,28,34,27,33,35,39)(25,30,31,32,29,37,36,38);(8,11,13,12)(9,15,16,21)(10,19,18,20)(14,17,22,23)364 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 12601);