Finite Group DatabasebetaFinite Group Databasebeta| order | 4736 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C37 · Schur multiplier (H2) = C22×C4 |
| direct product | C37×#1458 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C37 | C22⋊C4 |
| commutator | C22 | C22×C8×C37 |
| Frattini, cosocle | C22×C4 | C23×C37 |
| socle | C22×C37 | C2×C22⋊C4 |
| prime radical | C24×C37 | C2×C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C37 ◁ C2×C42×C37 ◁ G | |
264 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 264 conjugacy classes of subgroups — too many to draw. Only its 120 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) = 1184 |
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)(38,39,40,42,41,43,44,45)(46,47)(48,54,56,51)(49,52)(50,53)(55,60,61,58)(57,59);(46,48)(47,51)(49,55)(50,56)(52,58)(53,54)(57,61)(59,60);(46,49,50,57)(47,52,53,59)(48,55,56,61)(51,58,54,60)1628 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C37×#7188, C37×#7190, C37×#7252, … |
| maximal normal subgroup of | C37×#7188, C37×#7190, C37×#7252, … |
| maximal quotient of | C37×#7560, C37×#7562, C37×#7574, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4736, 190);