Finite Group DatabasebetaFinite Group Databasebeta| order | 4736 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C16×C37 · Schur multiplier (H2) = C22 |
| direct product | C37×D4⋊C16 |
| unambiguous semidirect product | D4⋊(C16×C37) |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C37 | D4 |
| commutator | C4 | C2×C16×C37 |
| Frattini, cosocle | C4×C8 | C22×C37 |
| socle | C22×C37 | C22⋊C8 |
| prime radical | C2×D4×C37 | C8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C37 ◁ C4×C8×C37 ◁ G | |
110 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 110 conjugacy classes of subgroups — too many to draw. Only its 56 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 61 |
| transitive degree | μt(G) = 2368 |
A minimal faithful permutation representation — 2 generators on 61 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)(40,42)(41,44)(43,45)(46,47,48,51,49,52,54,57,50,53,55,58,56,59,60,61);(39,42)(40,45)(43,44)2072 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C2×C37×D4⋊C16, C37×#11582, C37×#11583, … |
| maximal normal subgroup of | C2×C37×D4⋊C16, C37×#11582, C37×#11583, … |
| maximal quotient of | C37×#7068, C37×#7069, C37×#7070, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4736, 61);