Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13030 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×#13030 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | #1879 |
| commutator | C2×C8 | C22×C4×C7 |
| Frattini, cosocle | C22×C8 | C23×C7 |
| prime radical | C7×#3414 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C22×C4×C7 ◁ C7×#405 ◁ G | |
476 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3,4,5,6,7)(8,9,12,17,13,19,29,33,14,20,11,18,28,37,26,15)(10,21,23,34,24,36,30,35,25,32,22,39,31,38,27,16);(9,15,20,33)(11,26)(12,29)(13,28)(16,32,35,21)(17,19,18,37)(22,27)(23,30)(24,31)(34,36,39,38);(8,10)(9,16)(11,27)(12,30)(13,31)(14,25)(15,32)(17,36)(18,38)(19,39)(20,35)(21,33)(22,26)(23,29)(24,28)(34,37)280 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6018);