Finite Group DatabasebetaFinite Group Databasebeta| order | 4736 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #365 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C37 · Schur multiplier (H2) = C22 |
| direct product | C37×#2155 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C37 | #106 |
| commutator | C8 | C22×C4×C37 |
| Frattini, cosocle | C2×C8 | C23×C37 |
| socle | C2×C37 | C2×#106 |
| prime radical | C23×C37 | SD16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C37 ◁ C22×C4×C37 ◁ C22×C8×C37 ◁ G | |
144 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 144 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)(40,49,52,63)(41,46)(42,55)(43,58)(44,48)(45,51,60,65)(47,56)(50,62,53,67)(54,61)(57,68)(59,64,69,66);(38,40,42,51,43,53,56,66,44,52,57,65,41,50,54,64)(39,45,47,49,46,59,55,62,48,60,61,63,58,69,68,67);(38,41,44,43)(39,46,48,58)(40,50,52,53)(42,54,57,56)(45,59,60,69)(47,55,61,68)(49,62,63,67)(51,64,65,66)1184 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C37×#12668, C37×#12678, C37×#12832, … |
| maximal normal subgroup of | C37×#12668, C37×#12678, C37×#12832, … |
| maximal quotient of | C37×#11999, C37×#12000, C37×#12019, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4736, 887);