Finite Group DatabasebetaFinite Group Databasebeta| order | 4736 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C37 · Schur multiplier (H2) = C24 |
| direct product | C37×#2015 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C37 | C2×D4 |
| commutator, Frattini, cosocle | C22×C4 | C23×C37 |
| prime radical | C22×D4×C37 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C37 ◁ C2×C42×C37 ◁ G | |
390 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 390 conjugacy classes of subgroups — too many to draw. Only its 108 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)(38,39)(40,41)(46,47,51,55)(48,56,59,60)(49,54,50,53)(52,58,61,57);(39,41)(42,43)(44,45)(46,48,51,59)(47,52,55,61)(49,58,50,57)(53,60,54,56);(43,45)(47,53)(49,50)(52,60)(54,55)(56,61)(57,58)1184 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C2×C37×#2015, C37×#23421, C37×#23423, … |
| maximal normal subgroup of | C2×C37×#2015, C37×#23421, C37×#23423, … |
| maximal quotient of | C37×#9955, C37×#9956, C37×#9958, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4736, 747);