Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3019 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C7 · Schur multiplier (H2) = C27 |
| direct product | C7×#20939 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | C2×#120 |
| commutator | C2×C4 | C23×C4×C7 |
| Frattini, cosocle | C22×C4 | C24×C7 |
| socle | C2×C7 | C2×#386 |
| prime radical | C2×C7×#575 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C23×C4×C7 ◁ G | |
1668 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)(10,19,22,31)(11,23)(12,18)(13,27)(14,17)(15,21,30,26)(16,24)(20,35,38,29)(25,33)(28,32)(34,36,39,37);(8,10,12,26,14,22,13,21)(9,15,18,31,17,30,27,19)(11,20,24,37,25,38,28,36)(16,29,33,39,32,35,23,34);(8,11)(9,16)(10,20)(12,24)(13,28)(14,25)(15,29)(17,32)(18,33)(19,34)(21,36)(22,38)(23,27)(26,37)(30,35)(31,39);(9,17)(15,30)(16,32)(18,27)(19,31)(23,33)(29,35)(34,39)406 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 13927);