Finite Group DatabasebetaFinite Group Databasebeta| order | 4736 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2127 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C37 · Schur multiplier (H2) = C2×C42 |
| direct product | C37×#2128 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C37 | C2×#99 |
| commutator | C2×C4 | C22×C4×C37 |
| Frattini, cosocle | C2×D4 | C23×C37 |
| prime radical | C37×#535 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C37 ◁ C2×C4×C37 ◁ C2×#41×C37 ◁ G | |
270 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 270 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) = 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)(38,39)(40,44)(41,45,49,43)(42,46)(47,52)(48,51,53,50);(39,43)(40,47)(41,49)(44,51)(45,46)(50,52);(38,40,42,47)(39,44,46,52)(41,48,49,53)(43,50,45,51)851 conjugacy classes — view the full character table.
| maximal subgroup of | C37×#12711, C37×#12714, C37×#12722, … |
| maximal normal subgroup of | C37×#12711, C37×#12714, C37×#12722, … |
| maximal quotient of | C37×#11433, C37×#11450, C37×#11480, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4736, 860);