Finite Group DatabasebetaFinite Group Databasebeta| order | 1856 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C29 · Schur multiplier (H2) = C23 |
| direct product | C29×#452 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C29 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C23×C29 |
| prime radical | C2×#41×C29 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C29 ◁ C22×C4×C29 ◁ G | |
152 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 152 conjugacy classes of subgroups — too many to draw. Only its 58 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 49 |
| transitive degree | μt(G) = 928 |
A minimal faithful permutation representation — 3 generators on 49 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29)(30,31)(32,33)(34,35,39,42)(36,43,45,48)(37,41,38,46)(40,47,49,44);(34,36)(35,40)(37,44)(38,47)(39,45)(41,48)(42,49)(43,46);(31,33)(34,37,39,38)(35,41,42,46)(36,44,45,47)(40,48,49,43)551 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1856, 133);