Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3290 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C7 · Schur multiplier (H2) = C25 |
| direct product | C7×#23042 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | D42 |
| commutator | C2×C4 | C23×C4×C7 |
| Frattini, cosocle | C22×C4 | C24×C7 |
| socle | C2×C7 | #2361 |
| prime radical | C7×#3067 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C22×C4×C7 ◁ G | |
824 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)(8,9,15,19)(10,20)(11,25,27,12)(13,18,14,31)(16,30)(17,32,29,26)(21,22)(23,36)(24,35)(28,38)(33,39)(34,37);(8,10,14,23,15,24,13,30)(9,16,31,20,19,36,18,35)(11,21,32,38,27,39,26,34)(12,28,17,33,25,37,29,22);(8,11)(9,17)(10,21)(12,18)(13,26)(14,32)(15,27)(16,33)(19,29)(20,37)(22,36)(23,38)(24,39)(25,31)(28,35)(30,34);(8,12,13,29,15,25,14,17)(9,11,18,26,19,27,31,32)(10,22,23,28,24,33,30,37)(16,34,20,21,36,38,35,39)406 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 16030);