Finite Group DatabasebetaFinite Group Databasebeta| order | 4736 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C37 · Schur multiplier (H2) = C23 |
| direct product | C37×#1858 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C37 | #120 |
| commutator | C2×C4 | C22×C4×C37 |
| Frattini, cosocle | C22×C4 | C23×C37 |
| socle | C2×C37 | #385 |
| prime radical | C2×#41×C37 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C37 ◁ C22×C4×C37 ◁ G | |
252 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 252 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)(38,39,43,47,44,48,42,55)(40,49,51,45,52,60,56,59)(41,46,53,62,54,63,57,61)(50,65,68,69,67,64,66,58);(38,40,43,51,44,52,42,56)(39,45,47,60,48,59,55,49)(41,50,53,68,54,67,57,66)(46,58,62,65,63,69,61,64);(38,41)(39,46)(40,50)(42,57)(43,53)(44,54)(45,58)(47,62)(48,63)(49,64)(51,68)(52,67)(55,61)(56,66)(59,69)(60,65)1184 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C37×#7377, C37×#7378, C37×#12962, … |
| maximal normal subgroup of | C37×#7377, C37×#7378, C37×#12962, … |
| maximal quotient of | C37×#8835, C37×#8836, C37×#8837, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4736, 590);