Finite Group DatabasebetaFinite Group Databasebeta| order | 4736 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C37 · Schur multiplier (H2) = C4 |
| direct product | C37×#1572 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C37 | C2×D4 |
| commutator | C2×C4 | C22×C4×C37 |
| Frattini, cosocle | C42 | C23×C37 |
| socle | C22×C37 | C2×C4⋊C4 |
| prime radical | C23×C37 | C4⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C37 ◁ C2×C42×C37 ◁ G | |
150 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 150 conjugacy classes of subgroups — too many to draw. Only its 92 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) = 2368 |
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)(40,47)(41,45)(42,51)(43,46)(44,49)(48,52)(50,53)(54,55,57,60,58,61,64,67)(56,59,63,66,62,65,68,69);(38,40,42,49,43,50,41,48)(39,44,45,47,46,52,51,53)(54,56)(55,59)(57,63)(58,62)(60,66)(61,65)(64,68)(67,69);(38,41,43,42)(39,45,46,51)(40,48,50,49)(44,47,52,53)(56,62)(59,65)(63,68)(66,69)1406 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C2×C37×#1572, C37×#17803, C37×#17808, … |
| maximal normal subgroup of | C2×C37×#1572, C37×#17803, C37×#17808, … |
| maximal quotient of | C37×#7878, C37×#8277, C37×#8278, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4736, 304);