Finite Group DatabasebetaFinite Group Databasebeta| order | 4736 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C37 · Schur multiplier (H2) = C22 |
| direct product | C37×#1394 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C37 | #99 |
| commutator | C2×C4 | C42×C37 |
| Frattini, cosocle | C22×Q8 | C22×C37 |
| prime radical | C23×C37 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C37 ◁ C23×C37 ◁ C22×Q8×C37 ◁ G | |
172 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 172 conjugacy classes of subgroups — too many to draw. Only its 64 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 — 2 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)(42,43,45,49,46,50,54,57)(44,48,51,56,52,47,53,55);(38,39,40,41)(43,47,50,48)(49,56,57,55)962 conjugacy classes — view the full character table.
| maximal subgroup of | C37×#7404, C37×#7405, C37×#7409, … |
| maximal normal subgroup of | C37×#7404, C37×#7405, C37×#7409, … |
| maximal quotient of | C37×#7189, C37×#7206, C37×#7215, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4736, 126);