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×#1990 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C37 | C2×D4 |
| commutator | C2×C4 | C22×C4×C37 |
| Frattini, cosocle | C22×C4 | C23×C37 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C37 ◁ C2×C42×C37 ◁ G | |
300 conjugacy classes of subgroups; only the 160 normal subgroups are shown.
This group has 300 conjugacy classes of subgroups — too many to draw. Only its 160 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) = 4736 |
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)(44,48)(45,46)(47,49)(51,54)(52,56)(55,57);(39,41)(43,46)(47,48)(50,51,52,54,53,55,56,57);(42,44,45,49)(43,47,46,48)1628 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C37×#8446, C37×#8447, C37×#8448, … |
| maximal normal subgroup of | C37×#8446, C37×#8447, C37×#8448, … |
| maximal quotient of | C37×#9870, C37×#9874, C37×#9884, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4736, 722);