Finite Group DatabasebetaFinite Group Databasebeta| order | 4736 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3290 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C37 · Schur multiplier (H2) = C25 |
| direct product | C37×#3291 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C37 | D42 |
| commutator, Frattini, cosocle | C2×C4 | C24×C37 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C37 ◁ C2×C4×C37 ◁ G | |
460 conjugacy classes of subgroups; only the 184 normal subgroups are shown.
This group has 460 conjugacy classes of subgroups — too many to draw. Only its 184 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 — 4 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)(38,39,44,49)(40,50,51,52)(41,46,45,53)(42,48,43,47);(38,40)(39,45)(41,49)(42,53)(43,46)(44,51)(47,52)(48,50);(38,41)(39,40)(42,52)(43,50)(44,45)(46,47)(48,53)(49,51);(38,41)(39,46)(40,47)(42,50)(43,52)(44,45)(48,51)(49,53)1073 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C37×#12314, C37×#12315, C37×#12364, … |
| maximal normal subgroup of | C37×#12314, C37×#12315, C37×#12364, … |
| maximal quotient of | C37×#19884, C37×#19888, C37×#19890, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4736, 2023);