Finite Group DatabasebetaFinite Group Databasebeta| order | 4736 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C37 · Schur multiplier (H2) = C24 |
| direct product | C37×#1884 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C37 | #120 |
| commutator | C2×C4 | C22×C4×C37 |
| Frattini, cosocle | C22×C4 | C23×C37 |
| socle | C2×C37 | #386 |
| prime radical | C37×#532 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C37 ◁ C22×C4×C37 ◁ G | |
332 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 332 conjugacy classes of subgroups — too many to draw. Only its 108 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) = 592 |
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)(38,39)(40,47,48,51)(41,44,49,50)(42,45)(43,46)(52,53);(38,40,42,50,43,48,52,44)(39,41,45,51,46,49,53,47);(38,41)(39,44)(40,45)(42,51)(43,49)(46,50)(47,52)(48,53)1184 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C37×#7482, C37×#7484, C37×#7486, … |
| maximal normal subgroup of | C37×#7482, C37×#7484, C37×#7486, … |
| maximal quotient of | C37×#8957, C37×#8958, C37×#8959, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4736, 616);