Finite Group DatabasebetaFinite Group Databasebeta| order | 4736 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1665 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C37 · Schur multiplier (H2) = C22 |
| direct product | C37×#1668 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C37 | #121 |
| commutator, Frattini, cosocle | C42 | C23×C37 |
| prime radical | C37×#493 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C37 ◁ C42×C37 ◁ G | |
188 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 188 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) = 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)(39,42)(40,43)(44,45)(47,50)(48,52)(51,53);(38,39,43,45,41,44,40,42);(46,47,49,51)(48,53,52,50)1073 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C37×#12328, C37×#12329, C37×#12330, … |
| maximal normal subgroup of | C37×#12328, C37×#12329, C37×#12330, … |
| maximal quotient of | C37×#8645, C37×#9222, C37×#9564, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4736, 400);