Finite Group DatabasebetaFinite Group Databasebeta| order | 4736 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1665 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C37 · Schur multiplier (H2) = C2 |
| direct product | C37×#1684 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C37 | #121 |
| commutator, Frattini, cosocle | C42 | C23×C37 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C37 ◁ C42×C37 ◁ G | |
144 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 144 conjugacy classes of subgroups — too many to draw. Only its 68 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) = 4736 |
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)(38,39,42,45)(40,46,49,53)(41,50,47,44)(43,51,52,48)(54,55)(56,62)(57,59)(58,66)(60,64)(61,65)(63,69)(67,68);(38,40,47,51,42,49,41,48)(39,43,50,53,45,52,44,46)(55,59)(60,68)(61,66)(62,69);(54,56,57,63)(55,60,59,68)(58,67,65,64)(61,69,66,62)1073 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C2×C37×#1684, C37×#19123, C37×#19126, … |
| maximal normal subgroup of | C2×C37×#1684, C37×#19123, C37×#19126, … |
| maximal quotient of | C37×#8649, C37×#9216, C37×#9220, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4736, 416);