Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11867 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C23 |
| direct product | C7×#11867 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | #1519 |
| commutator | C2×C8 | C22×C4×C7 |
| Frattini, cosocle | C2×M16 | C23×C7 |
| prime radical | C7×#3060 | C2 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C2×C4×C7 ◁ C2×C7×#131 ◁ G | |
376 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)(10,20)(11,24,28,14)(12,17)(13,19)(15,21)(16,25,30,38)(18,27)(22,32,26,31)(23,33)(29,39,35,36)(34,37);(9,14)(10,21)(11,25)(12,27)(15,31)(16,18)(17,30)(19,24)(20,35)(22,26)(23,37)(28,38)(29,33)(32,34);(8,10,12,21,13,23,27,37)(9,15,18,20,19,34,17,33)(11,26,25,39,28,22,38,36)(14,29,30,32,24,35,16,31)238 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 4855);