Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12826 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C23 |
| direct product | C7×#12826 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | #1804 |
| commutator | C2×C8 | C22×C4×C7 |
| Frattini, cosocle | C22×C8 | C23×C7 |
| prime radical | C22×C7×D8 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C22×C4×C7 ◁ C7×#405 ◁ G | |
412 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)(10,20,23,37)(11,17)(12,16)(13,29)(14,19)(15,21,32,24)(18,27)(22,35,26,31)(25,33)(28,34)(30,39,36,38);(8,10)(9,15,19,32)(11,21)(12,26)(13,24)(14,23)(16,30,34,36)(17,20,29,37)(18,35,33,31)(22,28)(25,39)(27,38);(9,16)(11,13)(12,27)(15,30)(17,18)(19,34)(20,35)(21,24)(22,39)(25,28)(26,38)(29,33)(31,37)(32,36)280 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 5814);