Finite Group DatabasebetaFinite Group Databasebeta| order | 4736 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C37 · Schur multiplier (H2) = C24×C4 |
| direct product | C37×#3150 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C37 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C37 |
| socle | C22×C37 | C22×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C37 ◁ C2×C42×C37 ◁ G | |
564 conjugacy classes of subgroups; only the 216 normal subgroups are shown.
This group has 564 conjugacy classes of subgroups — too many to draw. Only its 216 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 — 4 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)(39,41)(42,43)(44,51)(45,49)(46,55)(47,50)(48,54)(52,57)(53,56);(42,44,46,54,47,52,45,53)(43,48,49,51,50,56,55,57);(38,39,40,41)(44,52)(48,56)(51,57)(53,54);(42,45,47,46)(43,49,50,55)(44,53,52,54)(48,51,56,57)1184 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C37×#9007, C37×#10032, C37×#10034, … |
| maximal normal subgroup of | C37×#9007, C37×#10032, C37×#10034, … |
| maximal quotient of | C37×#19608, C37×#19613, C37×#19616, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4736, 1882);