Finite Group DatabasebetaFinite Group Databasebeta| order | 4736 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C37 · Schur multiplier (H2) = C22×C4 |
| direct product | C37×#1976 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C37 | #121 |
| commutator | C2×C4 | C22×C4×C37 |
| Frattini, cosocle | C22×C4 | C23×C37 |
| socle | C2×C37 | #390 |
| prime radical | C2×#135×C37 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C37 ◁ C22×C4×C37 ◁ G | |
284 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 284 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)(40,44,49,58)(41,51)(42,54)(43,46)(45,52)(47,55)(48,60,61,69)(50,64,63,57)(53,65)(56,67,68,62)(59,66);(38,40)(39,44,46,58)(41,48)(42,50)(43,49)(45,56,59,68)(47,57,54,64)(51,60,65,69)(52,62)(53,61)(55,63)(66,67);(38,41,42,52,43,53,55,66)(39,45,47,51,46,59,54,65)(40,48,50,62,49,61,63,67)(44,56,57,60,58,68,64,69)1184 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C37×#13016, C37×#13021, C37×#13030, … |
| maximal normal subgroup of | C37×#13016, C37×#13021, C37×#13030, … |
| maximal quotient of | C37×#9667, C37×#9668, C37×#9669, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4736, 708);