Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3288 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C7 · Schur multiplier (H2) = C25 |
| direct product | C7×#23043 |
| 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 | |
820 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)(8,9,14,19)(10,20)(11,12,29,32)(13,17,21,18)(15,25)(16,27,31,28)(22,33)(23,39)(24,35)(26,36)(30,34)(37,38);(8,10,21,24,14,26,13,25)(9,15,18,20,19,35,17,36)(11,22,28,38,29,39,27,34)(12,30,16,33,32,37,31,23);(8,11)(9,16)(10,22)(12,17)(13,27)(14,29)(15,33)(18,32)(19,31)(20,37)(21,28)(23,35)(24,38)(25,34)(26,39)(30,36);(8,12,13,31,14,32,21,16)(9,11,17,27,19,29,18,28)(10,23,24,30,26,33,25,37)(15,34,20,22,35,38,36,39)364 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 16031);