Finite Group DatabasebetaFinite Group Databasebeta| order | 4736 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C37 · Schur multiplier (H2) = C23 |
| direct product | C37×#1390 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C37 | #99 |
| commutator | C23 | C42×C37 |
| Frattini, cosocle | C22×D4 | C22×C37 |
| prime radical | C22×D4×C37 | C22 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C37 ◁ C23×C37 ◁ C22×D4×C37 ◁ G | |
234 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 234 conjugacy classes of subgroups — too many to draw. Only its 64 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 49 |
| transitive degree | μt(G) = 1184 |
A minimal faithful permutation representation — 2 generators on 49 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)(42,43)(44,48,49,46)(45,47);(38,39,40,41)(43,46)(47,48)962 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C37×#1390, C37×#12573, C37×#12574, … |
| maximal normal subgroup of | C2×C37×#1390, C37×#12573, C37×#12574, … |
| maximal quotient of | C37×#7187, C37×#7203, C37×#7204, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4736, 122);