Finite Group DatabasebetaFinite Group Databasebeta| order | 4736 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2081 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C37 · Schur multiplier (H2) = C22 |
| direct product | C37×#2081 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C37 | #397 |
| commutator, Frattini, cosocle | C22×C4 | C23×C37 |
| prime radical | C22×D4×C37 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C37 ◁ C22×C4×C37 ◁ G | |
204 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 204 conjugacy classes of subgroups — too many to draw. Only its 72 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)(38,39,42,44,43,47,41,46)(40,48,50,53,49,52,51,45);(39,44)(40,49)(41,42)(45,52)(46,47)(48,53);(38,40,41,50,43,49,42,51)(39,45,46,48,47,53,44,52)851 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C37×#2081, C37×#23652, C37×#24987, … |
| maximal normal subgroup of | C2×C37×#2081, C37×#23652, C37×#24987, … |
| maximal quotient of | C37×#10631, C37×#10633, C37×#10634, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4736, 813);