Finite Group DatabasebetaFinite Group Databasebeta| order | 4736 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C37 · Schur multiplier (H2) = C23 |
| direct product | C37×#2132 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C37 | #99 |
| commutator | C2×C4 | C22×C4×C37 |
| Frattini, cosocle | C2×Q8 | C23×C37 |
| socle | C2×C37 | C2×#99 |
| prime radical | C37×#532 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C37 ◁ C2×C4×C37 ◁ C2×#41×C37 ◁ G | |
250 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 250 conjugacy classes of subgroups — too many to draw. Only its 84 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 — 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)(38,39,42,47,43,48,40,45)(41,46,49,53,50,44,51,52);(39,44)(41,50)(45,52)(46,48)(47,53)(49,51);(38,40,43,42)(39,45,48,47)(41,49,50,51)(44,52,46,53)962 conjugacy classes — view the full character table.
| maximal subgroup of | C37×#7484, C37×#7488, C37×#7492, … |
| maximal normal subgroup of | C37×#7484, C37×#7488, C37×#7492, … |
| maximal quotient of | C37×#11431, C37×#11447, C37×#11475, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4736, 864);