Finite Group DatabasebetaFinite Group Databasebeta| order | 4736 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C37 · Schur multiplier (H2) = C24 |
| direct product | C37×#1873 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C37 | C2×D4 |
| commutator | C2×C4 | C22×C4×C37 |
| Frattini, cosocle | C22×C4 | C23×C37 |
| prime radical | C24×C37 | D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C37 ◁ C23×C4×C37 ◁ G | |
360 conjugacy classes of subgroups; only the 128 normal subgroups are shown.
This group has 360 conjugacy classes of subgroups — too many to draw. Only its 128 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 61 |
| transitive degree | μt(G) = 2368 |
A minimal faithful permutation representation — 3 generators on 61 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37)(39,41)(42,43)(44,45)(46,47,50,54,51,55,49,53)(48,56,57,52,59,60,58,61);(38,39,40,41)(46,48)(47,52)(49,57)(50,58)(51,59)(53,60)(54,56)(55,61);(43,45)(46,49,51,50)(47,53,55,54)(48,57,59,58)(52,60,61,56)1406 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C37×#8488, C2×C37×#1873, C37×#20908, … |
| maximal normal subgroup of | C37×#8488, C2×C37×#1873, C37×#20908, … |
| maximal quotient of | C37×#8938, C37×#8941, C37×#8974, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4736, 605);