Finite Group DatabasebetaFinite Group Databasebeta| order | 4736 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C37 · Schur multiplier (H2) = C24 |
| direct product | C37×#1887 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C37 | #120 |
| commutator | C2×C4 | C22×C4×C37 |
| Frattini, cosocle | C22×C4 | C23×C37 |
| prime radical | C2×#127×C37 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C37 ◁ C22×C4×C37 ◁ G | |
444 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 444 conjugacy classes of subgroups — too many to draw. Only its 112 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 49 |
| transitive degree | μt(G) = 592 |
A minimal faithful permutation representation — 3 generators on 49 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)(39,41)(42,43,47,49,46,44,45,48);(38,39,40,41)(42,44,46,43)(45,48,47,49);(45,47)(48,49)1184 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C37×#7485, C37×#7493, C37×#12759, … |
| maximal normal subgroup of | C37×#7485, C37×#7493, C37×#12759, … |
| maximal quotient of | C37×#8959, C37×#8963, C37×#9001, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4736, 619);