Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1909 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C7×C8 · Schur multiplier (H2) = C24 |
| direct product | C7×#12988 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | #386 |
| commutator | C2×C4 | C22×C7×C8 |
| Frattini, cosocle | C22×C8 | C23×C7 |
| socle | C2×C7 | #1869 |
| prime radical | C7×#532 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C22×C7×C8 ◁ G | |
466 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)(8,9,13,19,14,17,31,34,15,18,32,36,12,29,21,38)(10,20,24,30,25,35,39,11,23,27,37,26,22,16,33,28);(8,10,21,33,12,22,32,37,15,23,31,39,14,25,13,24)(9,16,34,30,29,27,19,28,18,35,38,26,17,20,36,11);(8,11)(9,10)(12,30)(13,20)(14,26)(15,28)(16,21)(17,25)(18,23)(19,37)(22,29)(24,36)(27,32)(31,35)(33,34)(38,39)406 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 5976);