Finite Group DatabasebetaFinite Group Databasebeta| order | 4736 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C37 · Schur multiplier (H2) = C25 |
| direct product | C37×#1791 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C37 | #120 |
| commutator | C23 | C22×C4×C37 |
| Frattini, cosocle | C22×C4 | C23×C37 |
| socle | C2×C37 | #379 |
| prime radical | C37×#585 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C37 ◁ C22×C4×C37 ◁ G | |
548 conjugacy classes of subgroups; only the 132 normal subgroups are shown.
This group has 548 conjugacy classes of subgroups — too many to draw. Only its 132 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 69 |
| transitive degree | μt(G) = 1184 |
A minimal faithful permutation representation — 3 generators on 69 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)(38,39,42,47,43,46,55,60)(40,48,51,61,50,44,54,57)(41,45,53,58,52,59,65,69)(49,56,64,67,63,62,66,68);(38,40)(39,44)(41,49)(42,54)(43,50)(45,56)(46,48)(47,61)(51,55)(52,63)(53,66)(57,60)(58,68)(59,62)(64,65)(67,69);(38,41)(39,45)(40,49)(42,53)(43,52)(44,56)(46,59)(47,58)(48,62)(50,63)(51,64)(54,66)(55,65)(57,67)(60,69)(61,68)1184 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C37×#13045, C37×#13046, C37×#13051, … |
| maximal normal subgroup of | C37×#13045, C37×#13046, C37×#13051, … |
| maximal quotient of | C37×#8316, C37×#8317, C37×#8348, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4736, 523);