Finite Group DatabasebetaFinite Group Databasebeta| order | 4736 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16×C37 · Schur multiplier (H2) = C23 |
| direct product | C2×C37×C4⋊C16 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C8×C37 | C22 |
| commutator | C2 | C22×C16×C37 |
| Frattini, cosocle | C2×C8 | C23×C37 |
| socle, prime radical | C23×C37 | C2×C8 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C8×C37 ◁ G | |
184 conjugacy classes of subgroups; only the 152 normal subgroups are shown.
This group has 184 conjugacy classes of subgroups — too many to draw. Only its 152 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 59 |
| transitive degree | μt(G) = 4736 |
A minimal faithful permutation representation — 3 generators on 59 points:
(1,2)(23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59);(4,6)(7,8,9,12,10,13,15,18,11,14,16,19,17,20,21,22);(3,4,5,6)2960 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C37×#7151, C37×#7153, C37×#7167, … |
| maximal normal subgroup of | C37×#7151, C37×#7153, C37×#7167, … |
| maximal quotient of | C2×C37×#1367, C2×C37×#1370, C37×#12012, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4736, 881);