Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11521 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C22×C4 |
| direct product | C7×#12807 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | #410 |
| commutator | C22×C4 | C22×C4×C7 |
| Frattini, cosocle | C2×#41 | C23×C7 |
| socle | C2×C7 | #1799 |
| prime radical | C7×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C2×C7×C8 ◁ C2×C7×#131 ◁ G | |
428 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)(10,19)(11,23)(12,17,26,15)(13,18)(14,20)(16,25)(21,33,37,32)(22,34)(24,30,38,31)(27,35,28,39)(29,36);(8,10,11,20,13,22,25,36)(9,14,16,19,18,29,23,34)(12,21,24,35,26,37,38,39)(15,27,30,32,17,28,31,33);(9,15)(12,26)(14,27)(16,30)(17,18)(19,32)(21,37)(23,31)(24,38)(28,29)(33,34)(35,39)280 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 5795);