Finite Group DatabasebetaFinite Group Databasebeta| order | 4736 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2048 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C37 · Schur multiplier (H2) = C23 |
| direct product | C37×#2048 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C37 | #394 |
| commutator, Frattini, cosocle | C22×C4 | C23×C37 |
| prime radical | C2×AGL(1,ℤ/8)×C37 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C37 ◁ C22×C4×C37 ◁ G | |
274 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 274 conjugacy classes of subgroups — too many to draw. Only its 80 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)(39,44)(40,47)(41,42)(45,52)(46,50)(48,49);(38,39)(40,45)(41,50)(42,44)(43,46)(47,51)(48,53)(49,52);(38,40,41,48,43,49,42,47)(39,45,44,51,46,52,50,53)851 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C37×#2048, C37×#23525, C37×#24295, … |
| maximal normal subgroup of | C2×C37×#2048, C37×#23525, C37×#24295, … |
| maximal quotient of | C37×#10440, C37×#10441, C37×#10442, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4736, 780);