Finite Group DatabasebetaFinite Group Databasebeta| order | 4736 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C37 · Schur multiplier (H2) = C24 |
| direct product | C37×#1881 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C37 | #120 |
| commutator | C2×C4 | C22×C4×C37 |
| Frattini, cosocle | C22×C4 | C23×C37 |
| socle | C2×C37 | #386 |
| prime radical | C37×#575 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C37 ◁ C22×C4×C37 ◁ G | |
310 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 310 conjugacy classes of subgroups — too many to draw. Only its 108 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 69 |
| transitive degree | μt(G) = 1184 |
A minimal faithful permutation representation — 3 generators on 69 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)(40,49,52,60)(41,53)(42,47)(43,57)(44,48)(45,51,61,56)(46,54)(50,65,68,59)(55,62)(58,63)(64,66,69,67);(38,40,42,56,44,52,43,51)(39,45,47,60,48,61,57,49)(41,50,54,67,55,68,58,66)(46,59,62,69,63,65,53,64);(38,41)(39,46)(40,50)(42,54)(43,58)(44,55)(45,59)(47,62)(48,63)(49,64)(51,66)(52,68)(53,57)(56,67)(60,69)(61,65)1184 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C2×C37×#1881, C37×#20939, C37×#21477, … |
| maximal normal subgroup of | C2×C37×#1881, C37×#20939, C37×#21477, … |
| maximal quotient of | C37×#8953, C37×#8954, C37×#8955, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4736, 613);