Finite Group DatabasebetaFinite Group Databasebeta| order | 4736 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C37 · Schur multiplier (H2) = C24 |
| direct product | C37×#2041 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C37 | C2×D4 |
| commutator, Frattini, cosocle | C22×C4 | C23×C37 |
| prime radical | C2×#127×C37 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C37 ◁ C2×C42×C37 ◁ G | |
402 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 402 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) = 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)(38,39)(40,41)(43,44)(46,47)(48,50)(49,52)(51,53);(42,43)(44,45)(46,47)(48,50)(49,52)(51,53);(39,41)(42,44,45,43)(47,50)(48,53)(51,52)1184 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C37×#7735, C37×#7736, C37×#7737, … |
| maximal normal subgroup of | C37×#7735, C37×#7736, C37×#7737, … |
| maximal quotient of | C37×#10402, C37×#10403, C37×#10408, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4736, 773);