Finite Group DatabasebetaFinite Group Databasebeta| order | 4736 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C37 · Schur multiplier (H2) = C22 |
| direct product | C37×#1953 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C37 | #121 |
| commutator | C2×C4 | C22×C4×C37 |
| Frattini, cosocle | C22×C4 | C23×C37 |
| socle | C2×C37 | #389 |
| prime radical | C23×C37 | C2×Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C37 ◁ C22×C4×C37 ◁ G | |
184 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 184 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)(38,39,43,47,44,48,42,58)(40,49,51,45,53,60,52,61)(41,54,56,62,57,68,55,46)(50,59,64,67,66,69,65,63);(38,40,43,51,44,53,42,52)(39,45,47,60,48,61,58,49)(41,50,56,64,57,66,55,65)(46,59,54,67,62,69,68,63);(38,41,42,55,44,57,43,56)(39,46,47,54,48,62,58,68)(40,50,52,65,53,66,51,64)(45,59,60,67,61,69,49,63)1184 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C37×#12942, C37×#12947, C37×#12949, … |
| maximal normal subgroup of | C37×#12942, C37×#12947, C37×#12949, … |
| maximal quotient of | C37×#9607, C37×#9608, C37×#9609, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4736, 685);