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×#1986 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C37 | C2×D4 |
| commutator | C2×C4 | C22×C4×C37 |
| Frattini, cosocle | C22×C4 | C23×C37 |
| socle | C22×C37 | C2×C22⋊C4 |
| prime radical | C23×C37 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C37 ◁ C2×C42×C37 ◁ G | |
260 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 260 conjugacy classes of subgroups — too many to draw. Only its 116 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) = 1184 |
A minimal faithful permutation representation — 3 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)(40,44)(41,42)(43,45)(47,50,51,52);(39,42)(43,44)(46,47,48,52,49,51,53,50);(38,40,41,45)(39,43,42,44)1406 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C37×#8417, C37×#8418, C37×#8511, … |
| maximal normal subgroup of | C37×#8417, C37×#8418, C37×#8511, … |
| maximal quotient of | C37×#9864, C37×#9865, C37×#9866, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4736, 718);