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×#21014 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | #386 |
| commutator | C23 | C23×C4×C7 |
| Frattini, cosocle | C24 | 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 | |
1748 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)(10,14,20,27)(11,21,23,15)(12,16)(13,17)(18,25)(19,26,32,37)(22,34,35,28)(24,29)(30,36)(31,33)(38,39);(8,10)(9,14,17,27)(11,18)(12,19)(13,20)(15,25,21,33)(16,26,29,37)(22,30)(23,31)(24,32)(28,36,34,39)(35,38);(8,11)(9,15)(10,18)(12,22)(13,23)(14,25)(16,28)(17,21)(19,30)(20,31)(24,35)(26,36)(27,33)(29,34)(32,38)(37,39);(8,12,13,24)(9,16,17,29)(10,19,20,32)(11,22,23,35)(14,26,27,37)(15,28,21,34)(18,30,31,38)(25,36,33,39)406 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 14002);