Finite Group DatabasebetaFinite Group Databasebeta| order | 4736 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1404 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C37 · Schur multiplier (H2) = C2 |
| direct product | C37×#1405 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C37 | AΣL(1,𝔽16) |
| commutator | C2×Q8 | C2×C4×C37 |
| Frattini, cosocle | C4⋊Q8 | C22×C37 |
| prime radical | 2−1+4×C37 | C4 |
| LCS | G ▷ C2×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C37 ◁ C22×C37 ◁ C2×C4×C37 ◁ C4⋊Q8×C37 ◁ G | |
106 conjugacy classes of subgroups; only the 28 normal subgroups are shown.
This group has 106 conjugacy classes of subgroups — too many to draw. Only its 28 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 53 |
| transitive degree | μt(G) = 592 |
A minimal faithful permutation representation — 2 generators on 53 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)(38,39,41,45,43,40,47,49)(42,46,52,44,53,50,51,48);(38,40)(39,43)(41,44)(42,46)(45,52)(47,48)(49,51)(50,53)629 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C37×#1405, C37×#13572, C37×#13573, … |
| maximal normal subgroup of | C2×C37×#1405, C37×#13572, C37×#13573, … |
| maximal quotient of | C37×#7345, C37×#7376, C37×#7409, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4736, 137);