Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12753 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C25 |
| direct product | C7×#12758 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | #1790 |
| commutator | C2×D4 | C22×C4×C7 |
| Frattini, cosocle | C2×#41 | C23×C7 |
| prime radical | C7×#3540 | C2 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C2×C4×C7 ◁ C2×C7×#41 ◁ G | |
758 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)(8,9,13,18,14,19,11,16)(10,20,24,35,23,36,21,37)(12,26,25,33,27,17,28,32)(15,29,30,22,31,38,34,39);(8,10,14,23)(9,15,19,31)(11,21,13,24)(12,22,27,39)(16,30,18,34)(17,20,26,36)(25,38,28,29)(32,37,33,35);(8,11,14,13)(9,16,19,18)(10,21,23,24)(12,25,27,28)(15,30,31,34)(17,33,26,32)(20,35,36,37)(22,38,39,29)238 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 5746);