Finite Group DatabasebetaFinite Group Databasebeta| order | 4736 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C37 · Schur multiplier (H2) = C24 |
| direct product | Q8×SD16×C37 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C37 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C37 |
| prime radical | C2×D4×C37 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C37 ◁ C4×Q8×C37 ◁ G | |
370 conjugacy classes of subgroups; only the 208 normal subgroups are shown.
This group has 370 conjugacy classes of subgroups — too many to draw. Only its 208 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 53 |
| transitive degree | μt(G) = 2368 |
A minimal faithful permutation representation — 4 generators on 53 points:
(1,2,3,4)(5,8,7,6)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53);(1,5,3,7)(2,6,4,8);(9,10,11,12,13,14,15,16);(10,12)(11,15)(14,16)1295 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C37×#19765, C37×#19769, C37×#19796, … |
| maximal normal subgroup of | C37×#19765, C37×#19769, C37×#19796, … |
| maximal quotient of | Q8×#102×C37, Q8×#103×C37, C4⋊C4×SD16×C37, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4736, 2111);