Finite Group DatabasebetaFinite Group Databasebeta| order | 4736 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C37 · Schur multiplier (H2) = C2×C4 |
| direct product | C37×#1991 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C37 | C2×D4 |
| commutator | C2×C4 | C22×C4×C37 |
| Frattini, cosocle | C22×C4 | C23×C37 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C37 ◁ C2×C42×C37 ◁ G | |
252 conjugacy classes of subgroups; only the 128 normal subgroups are shown.
This group has 252 conjugacy classes of subgroups — too many to draw. Only its 128 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 65 |
| transitive degree | μt(G) = 4736 |
A minimal faithful permutation representation — 3 generators on 65 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)(38,39,40,41)(42,43)(44,48)(45,46)(47,49)(50,51)(52,58)(53,62)(54,56)(55,60)(57,59)(61,64)(63,65);(39,41)(43,46)(47,48)(50,52,59,65,54,61,53,60)(51,55,62,64,56,63,57,58);(42,44,45,49)(43,47,46,48)(51,56)(55,63)(57,62)(58,64)1628 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C37×#7135, C37×#7359, C37×#8529, … |
| maximal normal subgroup of | C37×#7135, C37×#7359, C37×#8529, … |
| maximal quotient of | C37×#9871, C37×#9875, C37×#9886, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4736, 723);