Finite Group DatabasebetaFinite Group Databasebeta| order | 4736 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C37 · Schur multiplier (H2) = C23 |
| direct product | C37×#1811 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C37 | C2×D4 |
| commutator | C2×C4 | C22×C4×C37 |
| Frattini, cosocle | C22×C4 | C23×C37 |
| socle | C22×C37 | C2×C4⋊C4 |
| prime radical | C24×C37 | Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C37 ◁ C23×C4×C37 ◁ G | |
272 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 272 conjugacy classes of subgroups — too many to draw. Only its 120 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) = 1184 |
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)(42,43,46,49,47,50,45,55)(44,51,54,56,52,57,53,48);(42,44,45,53,47,52,46,54)(43,48,49,51,50,56,55,57);(39,41)(44,52)(48,56)(51,57)(53,54)1184 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C37×#7256, C37×#7257, C37×#12740, … |
| maximal normal subgroup of | C37×#7256, C37×#7257, C37×#12740, … |
| maximal quotient of | C37×#8597, C37×#8598, C37×#8599, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4736, 543);