Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #365 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C7×C8 · Schur multiplier (H2) = C22 |
| direct product | C7×#7180 |
| subgroup | G/H | |
|---|---|---|
| center | C7×C8 | #106 |
| commutator | C8 | C4×C7×C8 |
| Frattini, cosocle | C2×C4×C8 | C22×C7 |
| socle | C2×C7 | #1279 |
| prime radical | C23×C7 | #104 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C7×C8 ◁ C22×C7×C8 ◁ C2×C4×C7×C8 ◁ G | |
240 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 2 generators on 39 points:
(1,2,3,4,5,6,7)(8,9,12,18,14,20,30,35,15,21,29,17,13,19,11,28)(10,22,24,33,26,38,23,34,27,36,31,32,25,37,39,16);(8,10)(9,16,19,32,21,34,20,33)(11,23)(12,31)(13,25)(14,26)(15,27)(17,22,35,37,18,36,28,38)(24,29)(30,39)448 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 168);