Finite Group DatabasebetaFinite Group Databasebeta| order | 4736 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C37 · Schur multiplier (H2) = C22 |
| direct product | C37×#1579 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C37 | C2×D4 |
| commutator | C4 | C22×C8×C37 |
| Frattini, cosocle | C42 | C23×C37 |
| socle, prime radical | C22×C37 | C4×D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C37 ◁ C4×C8×C37 ◁ G | |
160 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 160 conjugacy classes of subgroups — too many to draw. Only its 100 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)(38,39,41,44,42,40,43,45)(46,47)(48,54)(49,57)(50,52)(51,55)(53,59)(56,61)(58,60);(46,48,50,56)(47,51,52,60)(49,58,59,55)(53,61,57,54);(38,40,41,45,42,39,43,44)(47,52)(51,60)(53,57)(54,61)1628 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C37×#12065, C37×#12066, C37×#12079, … |
| maximal normal subgroup of | C37×#12065, C37×#12066, C37×#12079, … |
| maximal quotient of | C37×#8201, C37×#8284, C37×#8846, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4736, 311);