Finite Group DatabasebetaFinite Group Databasebeta| order | 4736 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1655 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C37 · Schur multiplier (H2) = C22 |
| direct product | C37×#1660 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C37 | #120 |
| commutator, Frattini, cosocle | C42 | C23×C37 |
| prime radical | C2×D4×C37 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C37 ◁ C42×C37 ◁ G | |
196 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 196 conjugacy classes of subgroups — too many to draw. Only its 60 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)(39,42)(40,44)(43,45)(46,47,52,56)(48,57,59,60)(49,58,53,61)(50,55,51,54);(40,44)(42,45)(46,48)(47,53)(49,56)(50,61)(51,58)(52,59)(54,60)(55,57);(38,39)(40,45)(41,43)(42,44)(46,49,52,53)(47,48,56,59)(50,60,51,57)(54,58,55,61)740 conjugacy classes — view the full character table.
| maximal subgroup of | C37×#13680, C2×C37×#1660, C37×#18984, … |
| maximal normal subgroup of | C37×#13680, C2×C37×#1660, C37×#18984, … |
| maximal quotient of | C37×#7751, C37×#7754, C37×#7757, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4736, 392);