Finite Group DatabasebetaFinite Group Databasebeta| order | 4736 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C37 · Schur multiplier (H2) = C24 |
| direct product | C37×#1459 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C37 | C22⋊C4 |
| commutator | C22 | C22×C8×C37 |
| Frattini, cosocle | C22×C4 | C23×C37 |
| socle | C22×C37 | C2×C22⋊C4 |
| prime radical | C22×D4×C37 | C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C37 ◁ C23×C4×C37 ◁ G | |
332 conjugacy classes of subgroups; only the 124 normal subgroups are shown.
This group has 332 conjugacy classes of subgroups — too many to draw. Only its 124 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)(40,44,45,42)(41,43)(46,47,49,52,50,53,48,51);(38,40)(39,42)(41,45)(43,44);(46,48,50,49)(47,51,53,52)1628 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C37×#7203, C37×#7205, C37×#7285, … |
| maximal normal subgroup of | C37×#7203, C37×#7205, C37×#7285, … |
| maximal quotient of | C37×#7565, C37×#7566, C37×#7582, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4736, 191);