Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1913 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C7 · Schur multiplier (H2) = C27 |
| direct product | C7×#21017 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | #386 |
| commutator | C23 | C23×C4×C7 |
| Frattini, cosocle | C22×C4 | C24×C7 |
| socle | C2×C7 | C2×#386 |
| prime radical | C7×#3449 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C23×C4×C7 ◁ G | |
1692 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)(8,9,13,19,14,18,12,17)(10,15,21,29,22,30,26,32)(11,23,24,33,25,16,27,31)(20,28,36,39,35,37,34,38);(8,10,13,26,14,22,12,21)(9,15,17,29,18,30,19,32)(11,20,27,36,25,35,24,34)(16,28,31,38,23,37,33,39);(8,11)(9,16)(10,20)(12,24)(13,27)(14,25)(15,28)(17,31)(18,23)(19,33)(21,34)(22,35)(26,36)(29,38)(30,37)(32,39);(8,12,14,13)(9,17,18,19)(10,21,22,26)(11,24,25,27)(15,29,30,32)(16,31,23,33)(20,34,35,36)(28,38,37,39)406 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 14005);