Finite Group DatabasebetaFinite Group Databasebeta| order | 4736 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C37 · Schur multiplier (H2) = C22×C4 |
| direct product | C37×#1958 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C37 | C2×D4 |
| commutator | C2×C4 | C22×C4×C37 |
| Frattini, cosocle | C22×C4 | C23×C37 |
| prime radical | C22×D4×C37 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C37 ◁ C2×C42×C37 ◁ G | |
336 conjugacy classes of subgroups; only the 128 normal subgroups are shown.
This group has 336 conjugacy classes of subgroups — too many to draw. Only its 128 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 61 |
| transitive degree | μt(G) = 2368 |
A minimal faithful permutation representation — 3 generators on 61 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)(39,41)(42,43,44,45)(46,47)(48,54)(49,57)(50,52)(51,55)(53,58)(56,60)(59,61);(43,45)(46,48,49,55,50,56,58,61)(47,51,53,54,52,59,57,60);(38,39,40,41)(47,52)(51,59)(53,57)(54,60)1628 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C37×#7502, C37×#8528, C2×C37×#1958, … |
| maximal normal subgroup of | C37×#7502, C37×#8528, C2×C37×#1958, … |
| maximal quotient of | C37×#9639, C37×#9645, C37×#9681, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4736, 690);