Finite Group DatabasebetaFinite Group Databasebeta| order | 4736 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C37 · Schur multiplier (H2) = C2×C4 |
| direct product | C37×#1930 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C37 | #121 |
| commutator | C2×C4 | C22×C4×C37 |
| Frattini, cosocle | C22×C4 | C23×C37 |
| prime radical | C23×C37 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C37 ◁ C22×C4×C37 ◁ G | |
212 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 212 conjugacy classes of subgroups — too many to draw. Only its 96 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)(38,39,41,40)(43,47)(44,50)(45,52)(48,55)(49,54)(53,56);(38,40,41,39)(42,43,45,47,46,49,52,54)(44,48,51,53,50,56,57,55);(42,44,45,51,46,50,52,57)(43,48,47,53,49,56,54,55)1184 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C2×C37×#1930, C37×#21075, C37×#21076, … |
| maximal normal subgroup of | C2×C37×#1930, C37×#21075, C37×#21076, … |
| maximal quotient of | C37×#9549, C37×#9551, C37×#9561, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4736, 662);