Finite Group DatabasebetaFinite Group Databasebeta| order | 4736 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C37 · Schur multiplier (H2) = C23 |
| direct product | C2×C37×#358 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C37 | D8 |
| commutator | C8 | C22×C4×C37 |
| Frattini, cosocle | C2×C8 | C23×C37 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C37 ◁ C22×C4×C37 ◁ C22×C8×C37 ◁ G | |
232 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 232 conjugacy classes of subgroups — too many to draw. Only its 120 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 59 |
| transitive degree | μt(G) = 4736 |
A minimal faithful permutation representation — 3 generators on 59 points:
(1,2)(23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59);(3,4,5,6)(8,12)(9,15)(10,17)(13,22)(14,21)(16,19)(18,20);(7,8,11,14)(9,16,18,22)(10,19,17,13)(12,20,21,15)1628 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C4×C37×#358, C37×#12638, C37×#12732, … |
| maximal normal subgroup of | C4×C37×#358, C37×#12638, C37×#12732, … |
| maximal quotient of | C2×C37×#1332, C37×#11646, C37×#11658, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4736, 869);