Finite Group DatabasebetaFinite Group Databasebeta| order | 4736 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1468 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C37 · Schur multiplier (H2) = C2×C4 |
| direct product | C37×#1473 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C37 | C2×C22⋊C4 |
| commutator | C23 | C22×C4×C37 |
| Frattini, cosocle | C22×C4 | C23×C37 |
| prime radical | C24×C37 | C2×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C37 ◁ C2×C42×C37 ◁ G | |
214 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 214 conjugacy classes of subgroups — too many to draw. Only its 88 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)(38,39,41,46,43,47,53,60)(40,45,50,58,48,59,61,69)(42,44,52,55,49,56,63,67)(51,54,65,66,62,57,64,68);(39,44)(40,48)(41,52)(45,57)(47,56)(50,64)(51,62)(53,63)(54,59)(58,69)(61,65)(66,68);(38,40,49,62)(39,45,56,57)(41,50,63,64)(42,51,43,48)(44,54,47,59)(46,58,67,68)(52,65,53,61)(55,66,60,69)1184 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C37×#8489, C37×#8490, C2×C37×#1473, … |
| maximal normal subgroup of | C37×#8489, C37×#8490, C2×C37×#1473, … |
| maximal quotient of | C37×#7666, C37×#7669, C37×#7672, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4736, 205);