Finite Group DatabasebetaFinite Group Databasebeta| order | 4736 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C37 · Schur multiplier (H2) = C23 |
| direct product | C37×#2036 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C37 | C2×D4 |
| commutator, Frattini, cosocle | C22×C4 | C23×C37 |
| prime radical | C24×C37 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C37 ◁ C23×C4×C37 ◁ G | |
316 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 316 conjugacy classes of subgroups — too many to draw. Only its 104 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 53 |
| transitive degree | μt(G) = 2368 |
A minimal faithful permutation representation — 3 generators on 53 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)(43,44)(46,47,48,50)(49,51,53,52);(38,39)(40,41)(42,43)(44,45)(47,50)(51,52);(39,41)(42,44,45,43)(47,51)(49,53)(50,52)1184 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C2×C37×#2036, C37×#23479, C37×#23481, … |
| maximal normal subgroup of | C2×C37×#2036, C37×#23479, C37×#23481, … |
| maximal quotient of | C37×#10399, C37×#10400, C37×#10413, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4736, 768);