Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1909 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C7 · Schur multiplier (H2) = C27 |
| direct product | C7×#21009 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | #386 |
| commutator | C2×C4 | C23×C4×C7 |
| Frattini, cosocle | C22×C4 | C24×C7 |
| socle | C2×C7 | C2×#386 |
| prime radical | C7×#3449 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C23×C4×C7 ◁ G | |
1708 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 4 generators on 39 points:
(1,2,3,4,5,6,7)(8,9)(10,19)(11,24,26,16)(12,17)(13,27)(14,18)(15,22)(20,33,36,39)(21,30)(23,34)(25,32,28,31)(29,35,38,37);(8,10,13,22,14,23,12,21)(9,15,17,19,18,30,27,34)(11,20,28,37,26,36,25,35)(16,29,31,33,24,38,32,39);(8,11)(9,16)(10,20)(12,25)(13,28)(14,26)(15,29)(17,31)(18,24)(19,33)(21,35)(22,37)(23,36)(27,32)(30,38)(34,39);(8,12,14,13)(9,17,18,27)(10,21,23,22)(11,25,26,28)(15,19,30,34)(16,31,24,32)(20,35,36,37)(29,33,38,39)406 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 13997);