Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C7×C8 · Schur multiplier (H2) = C2×C4 |
| direct product | C7×#8781 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | #117 |
| commutator | C2×C4 | C22×C7×C8 |
| Frattini, cosocle | C2×C42 | C23×C7 |
| socle | C22×C7 | #382 |
| prime radical | C23×C7 | #117 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C7×#376 ◁ G | |
310 conjugacy classes of subgroups.
| minimal degree | μ(G) = 47 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 3 generators on 47 points:
(1,2,3,4,5,6,7)(8,9,11,13,10,12,14,15)(16,17)(18,26,29,22)(19,30,32,45)(20,25)(21,24)(23,31,36,46)(27,40)(28,39,43,35)(33,37)(34,41)(38,47)(42,44);(9,12)(13,15)(16,18,20,28,21,29,33,43)(17,22,25,35,24,26,37,39)(19,27,31,41,32,42,46,47)(23,34,30,44,36,38,45,40);(16,19)(17,23,24,36)(18,27)(20,31)(21,32)(22,34,26,38)(25,30,37,45)(28,41)(29,42)(33,46)(35,44,39,40)(43,47)448 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 1769);