Finite Group DatabasebetaFinite Group Databasebeta| order | 4736 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1284 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C37 · Schur multiplier (H2) = C23 |
| direct product | C37×#1289 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C37 | #95 |
| commutator | C2×C4 | C42×C37 |
| Frattini, cosocle | C23×C4 | C22×C37 |
| prime radical | C24×C37 | D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C37 ◁ C23×C4×C37 ◁ G | |
228 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 228 conjugacy classes of subgroups — too many to draw. Only its 84 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) = 1184 |
A minimal faithful permutation representation — 2 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)(39,43)(40,41)(44,45)(46,47)(48,52,53,50)(49,51);(38,39,41,44,42,45,40,43)(46,48)(47,50)(49,53)(51,52)1184 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C37×#7076, C37×#7077, C37×#7078, … |
| maximal normal subgroup of | C37×#7076, C37×#7077, C37×#7078, … |
| maximal quotient of | C37×#7020, C37×#7024, C37×#7035, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4736, 21);