Finite Group DatabasebetaFinite Group Databasebeta| order | 4736 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C37 · Schur multiplier (H2) = C2 |
| direct product | C37×#1584 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C37 | C2×D4 |
| commutator | C2×C4 | C22×C4×C37 |
| Frattini, cosocle | C42 | C23×C37 |
| socle, prime radical | C22×C37 | C4×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C37 ◁ C4×C8×C37 ◁ G | |
152 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 152 conjugacy classes of subgroups — too many to draw. Only its 88 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) = 4736 |
A minimal faithful permutation representation — 3 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)(38,39,41,40,42,43,45,44)(46,47,50,53)(48,54,56,61)(49,57,59,52)(51,58,60,55);(46,48,50,56)(47,51,53,60)(49,58,59,55)(52,61,57,54);(38,40,41,43,42,44,45,39)1406 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C2×C37×#1584, C37×#17863, C37×#17870, … |
| maximal normal subgroup of | C2×C37×#1584, C37×#17863, C37×#17870, … |
| maximal quotient of | C37×#8203, C37×#8211, C37×#8291, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4736, 316);