Finite Group DatabasebetaFinite Group Databasebeta| order | 4736 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C37 · Schur multiplier (H2) = C24 |
| direct product | C37×#1908 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C37 | #121 |
| commutator | C2×C4 | C22×C4×C37 |
| Frattini, cosocle | C22×C4 | C23×C37 |
| prime radical | C2×#41×C37 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C37 ◁ C22×C4×C37 ◁ G | |
312 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 312 conjugacy classes of subgroups — too many to draw. Only its 112 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) = 2368 |
A minimal faithful permutation representation — 3 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)(42,43,47,49,46,45,53,54)(44,50,51,57,52,48,55,56);(42,44,47,55,46,52,53,51)(43,48,54,56,45,50,49,57);(39,41)(42,45)(43,46)(44,50)(47,49)(48,52)(51,56)(53,54)(55,57)1184 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C2×C37×#1908, C37×#21003, C37×#21004, … |
| maximal normal subgroup of | C2×C37×#1908, C37×#21003, C37×#21004, … |
| maximal quotient of | C37×#9096, C37×#9098, C37×#9165, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4736, 640);