Finite Group DatabasebetaFinite Group Databasebeta| order | 4736 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C37 · Schur multiplier (H2) = C2×C4 |
| direct product | C37×#1391 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C37 | #99 |
| commutator | C23 | C42×C37 |
| Frattini, cosocle | C2×#41 | C22×C37 |
| socle | C2×C37 | #342 |
| prime radical | C2×#41×C37 | C22 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C37 ◁ C2×C4×C37 ◁ C2×#41×C37 ◁ G | |
166 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 166 conjugacy classes of subgroups — too many to draw. Only its 60 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 69 |
| transitive degree | μt(G) = 1184 |
A minimal faithful permutation representation — 2 generators on 69 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)(40,48)(41,52,53,45)(42,46)(43,47)(44,49,56,62)(50,60)(51,61)(54,65,66,58)(55,59)(57,63,67,69)(64,68);(38,40,42,50,43,51,55,64)(39,44,46,57,47,56,59,67)(41,49,54,63,53,62,66,69)(45,48,58,60,52,61,65,68)962 conjugacy classes — view the full character table.
| maximal subgroup of | C37×#7398, C37×#7399, C37×#7406, … |
| maximal normal subgroup of | C37×#7398, C37×#7399, C37×#7406, … |
| maximal quotient of | C37×#7188, C37×#7204, C37×#7214, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4736, 123);