Finite Group DatabasebetaFinite Group Databasebeta| order | 4736 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C37 · Schur multiplier (H2) = C23 |
| direct product | C37×#2143 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C37 | C2×D8 |
| commutator | C8 | C22×C4×C37 |
| Frattini, cosocle | C2×C8 | C23×C37 |
| socle | C2×C37 | C2×#102 |
| prime radical | C2×#135×C37 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C37 ◁ C22×C4×C37 ◁ C22×C8×C37 ◁ G | |
220 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 220 conjugacy classes of subgroups — too many to draw. Only its 100 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)(38,39,42,47,44,48,43,58)(40,49,52,66,50,65,59,63)(41,53,55,64,57,68,56,46)(45,54,61,67,60,69,62,51);(38,40)(39,45)(41,54)(42,52)(43,59)(44,50)(46,63)(47,61)(48,60)(49,53)(51,56)(55,67)(57,69)(58,62)(64,66)(65,68);(40,50)(45,60)(49,65)(51,67)(52,59)(54,69)(61,62)(63,66)1184 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C37×#12649, C37×#12651, C37×#12740, … |
| maximal normal subgroup of | C37×#12649, C37×#12651, C37×#12740, … |
| maximal quotient of | C37×#11592, C37×#11593, C37×#11594, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4736, 875);