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 | C2×C37×#365 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C37 | #106 |
| commutator | C8 | C22×C4×C37 |
| Frattini, cosocle | C2×C8 | C23×C37 |
| prime radical | C23×C37 | SD16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C37 ◁ C22×C4×C37 ◁ C22×C8×C37 ◁ G | |
152 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 152 conjugacy classes of subgroups — too many to draw. Only its 104 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 55 |
| transitive degree | μt(G) = 1184 |
A minimal faithful permutation representation — 3 generators on 55 points:
(1,2)(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55);(4,8,9,14)(5,11)(6,13)(10,17,16,15)(12,18);(3,4,5,8,6,10,11,15,7,9,12,14,13,16,18,17)1184 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C4×C37×#365, C37×#12677, C37×#12831, … |
| maximal normal subgroup of | C4×C37×#365, C37×#12677, C37×#12831, … |
| maximal quotient of | C2×C37×#1368, C37×#12000, C37×#12019, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4736, 886);