Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C7×C8 · Schur multiplier (H2) = C4×C8 |
| direct product | C7×#8137 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C7×C8 | C42 |
| commutator | C4 | C2×C4×C7×C8 |
| Frattini, cosocle | C2×C42 | C23×C7 |
| socle | C22×C7 | #376 |
| prime radical | C23×C7 | #117 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C2×C7×C8 ◁ G | |
354 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)(8,9,10,12,11,13,14,15)(16,17)(18,26,29,39)(19,23)(20,24)(21,25)(22,28,34,42)(27,38,41,46)(30,35)(31,36)(32,37)(33,40,44,47)(43,45);(16,18)(17,22,25,34)(19,27)(20,28)(21,29)(23,33,36,44)(24,26,37,39)(30,40)(31,41)(32,42)(35,38,45,46)(43,47);(16,19,20,30,21,31,32,43)(17,23,24,35,25,36,37,45)(18,27,28,40,29,41,42,47)(22,33,26,38,34,44,39,46)616 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 1125);