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×#12828 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | #1804 |
| commutator | C2×C8 | C22×C4×C7 |
| Frattini, cosocle | C22×C8 | C23×C7 |
| prime radical | C2×C7×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C22×C4×C7 ◁ C7×#405 ◁ G | |
390 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)(8,9,11,17,14,19,13,31)(10,20,23,15,24,34,21,35)(12,26,25,32,30,39,29,18)(16,28,37,27,33,38,36,22);(8,10)(9,15,19,35)(11,21)(12,27)(13,23)(14,24)(16,32,33,18)(17,20,31,34)(22,30)(25,28)(26,36,39,37)(29,38);(8,10)(9,16)(11,23)(12,28)(13,21)(14,24)(15,32)(17,37)(18,35)(19,33)(20,26)(22,29)(25,27)(30,38)(31,36)(34,39)280 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 5816);