Finite Group DatabasebetaFinite Group Databasebeta| order | 4736 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C37 · Schur multiplier (H2) = C24 |
| direct product | C37×#2983 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C37 | C24 |
| commutator | C22 | C23×C4×C37 |
| Frattini, cosocle | C2×C4 | C24×C37 |
| socle | C22×C37 | C23×C4 |
| prime radical | C23×C37 | C22×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C37 ◁ G | |
356 conjugacy classes of subgroups; only the 268 normal subgroups are shown.
This group has 356 conjugacy classes of subgroups — too many to draw. Only its 268 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 — 4 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)(38,39,40,43,41,42,44,45)(46,47)(48,54)(49,52)(50,53)(51,56)(55,60)(57,59)(58,61);(46,48,50,56)(47,51,53,54)(49,57,58,55)(52,60,61,59);(39,42)(43,45)(46,49,50,58)(47,52,53,61)(48,55,56,57)(51,59,54,60);(47,53)(51,54)(52,61)(59,60)1628 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C37×#8838, C37×#8865, C37×#8962, … |
| maximal normal subgroup of | C37×#8838, C37×#8865, C37×#8962, … |
| maximal quotient of | C37×#20534, C37×#20536, C37×#20570, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4736, 1715);