Finite Group DatabasebetaFinite Group Databasebeta| order | 4736 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C37 · Schur multiplier (H2) = C22 |
| direct product | C37×#1386 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C37 | #102 |
| commutator | C8 | C42×C37 |
| Frattini, cosocle | C22×C8 | C22×C37 |
| prime radical | C23×C37 | D8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C37 ◁ C22×C4×C37 ◁ C22×C8×C37 ◁ G | |
140 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 140 conjugacy classes of subgroups — too many to draw. Only its 80 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 57 |
| transitive degree | μt(G) = 1184 |
A minimal faithful permutation representation — 2 generators on 57 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)(42,43,46,50)(44,45,52,54)(47,49,55,57)(48,56,53,51);(42,44,47,51,48,49,54,50,46,52,55,56,53,57,45,43)1184 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C2×C37×#1386, C37×#12567, C4×C37×#360, … |
| maximal normal subgroup of | C2×C37×#1386, C37×#12567, C4×C37×#360, … |
| maximal quotient of | C37×#7158, C37×#7175, C37×#7178, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4736, 118);